Download Preparing for the Regents Examination Geometry, AK

ANSWER KEY
Preparing for the
REGENTS EXAMINATION
GEOMETRY
AMSCO
A M S C O S C H O O L P U B L I C AT I O N S , I N C .
315 Hudson Street, New York, N.Y. 10013
N 81 CD
Compositor: Monotype LLC
Please visit our Web site at:
www.amscopub.com
Copyright © 2008 by Amsco School Publications, Inc.
No part of this book may be reproduced in any form without
written permission from the publisher.
Printed in the United States of America
1 2 3 4 5 6 7 8 9 10
12 11 10 09 08 07
Contents
Chapter 1: Essentials of Geometry
Chapter 2: Logic
Chapter 3: Introduction to Geometric Proof
Chapter 4: Congruence of Lines, Angles, and Triangles
Chapter 5: Congruence Based on Triangles
Chapter 6: Transformations and the Coordinate Plane
Chapter 7: Polygon Sides and Angles
Chapter 8: Slopes and Equations of Lines
Chapter 9: Parallel Lines
Chapter 10: Quadrilaterals
Chapter 11: Geometry of Three Dimensions
Chapter 12: Ratios, Proportion, and Similarity
Chapter 13: Geometry of the Circle
Chapter 14: Locus and Constructions
Cumulative Reviews
Practice Geometry Regents Examinations
1
3
8
12
18
28
34
39
44
56
68
72
79
87
92
138
CHAPTER
Essentials of Geometry
1-1 Undefined Terms
(page 2)
1 (1) finite
2 (2) infinite
3 (3) empty
4
A
B
5 A given line lies on an infinite number of
planes.
6 a and b
p
ᐉ
m
1-2 Real Numbers and
Their Properties
(pages 4–5)
1 (4) 8
2 (4) 5
3 (3) 0n 0
4 (1) a(b c) ab ac
5 (1) additive inverse
6 Commutative
7 Multiplicative identity
8 Commutative
9 Additive identity
10 Associative
11 Associative
12
13
14
15
16
17
18
19
20
1
Commutative
Distributive
Additive inverse
Distributive
Multiplicative inverse
1
0
0
Division by zero is undefined.
1-3 Lines and Line
Segments
(pages 6–7)
1 8
2 4
−−
−−
3 BD and CE
4 They must all lie on the same line.
5 They form a triangle.
6 Point S is the midpoint. It is halfway between P and T.
−−
7 Correct. The length of AB is equal to the
−−−
length of CD.
8 Incorrect notation
9 Incorrect notation
−−−
−−
10 Correct. AB is congruent to CD.
−−−
−−−
11 CPD bisects APB. It divides it into two segments with the same measure, 7.
12 MS 6
1-3 Lines and Line Segments
1
1-4 Angles
(pages 9–10)
1 (4) It is a portion of a line, beginning with
point M.
2 (1) DAE
3 (4) XYZ is a 180 angle.
4 (3) equal to 90
5 (2) three acute angles
6 Acute angles: JKM, MKN, NKO, OKL,
MKO
Obtuse angles: JKO, MKL
Right angles: JKN, NKL
Straight angle: JKL
7 mLKO 35; mMKN 30; mMKO 85
8
D
E
F
1-5 Preliminary Angle and
Line Relationships
(pages 11–12)
1 a mDBC 90
b DBC ABD
2 a mONP 45
b ONP MNO
3 mCBD
4 mCBD
5 mABD mABC mCBD
90 45
135
6 ln
7 x8
8 y 9.5
9 mCAT 28
10 mAFC 46
9 R
1-6 Triangles
T
B
10
V
11
G
H
I
12 a and b
A
E
I
B
C
(page 16)
1 False
2 True
3 False
4 False
5 True
6 The length of each leg of the isosceles
triangle is 18.
7 a The length of each side of the triangle
is 16.
b y7
8 mA 40; mB 70; mC 70
9 Check students’ drawings of isosceles
triangle ABC with vertex at C.
10 Check students’ drawings
___ of right triangle
PQR with hypotenuse PQ.
11 Check students’ drawings of acute triangle
FGH that is scalene.
−−−
−−
12 CD and CE
Chapter Review (pages 16–17)
1 Point
2 Line, length
3 Plane, length and width
4 9
5 Only the square roots of perfect square numbers are rational.
2
Chapter 1: Essentials of Geometry
−−
−−
6 a XZ YZ
b ZXY and ZYX
7 Coordinate of B is 2.
8 mKJL 36
−−
9 G is the midpoint of FH.
10 Bisectors pass through the same point, the
midpoint of the line segment. Distinct parallel lines have no points in common.
11
12
13
14
15
16
Straight angles: BCE and ACD
DEF
Right triangles: ABC and CED
CFE
−− −− −−
−−
BD, FD, FE, and BE are choices.
−−
BD
CHAPTER
2
Logic
2-1 Statements, Truth
Values, and Negations
(pages 20–21)
1 (2) a false closed sentence
2 (2) a false closed sentence
3 (4) not a statement
4 (1) a true closed sentence
5 (4) 4 3 is not greater than 2.
6 (2) The coat is not blue.
7 (2) false
8 (1) 3 6 7
9 Any open sentence
10 A triangle has three sides.
11 Statistics is the study of analyzing data.
12 Algebra is the study of operations on sets of
numbers.
13 Trigonometry is the study of triangles.
14 Geometry is the study of shapes and sizes.
15 “Chicago is not a city in Indiana.”
Original statement is false.
Negated statement is true.
16 “The sun does not rise in the west.”
Original statement is false.
Negated statement is true.
17 “January is not a winter month.”
Original statement is true.
Negated statement is false.
18 “The area of a rectangle is not length times
width.”
Original statement is true.
Negated statement is false.
19 “Each state does not have two senators.”
Original statement is true.
Negated statement is false.
20 “It is not the case that all real numbers are
rational.”
Original statement is false.
Negated statement is true.
2-2 Compound Statements
(pages 23–24)
1 (3) 3 7 10 or 4 7 3
2 (1) Albany is not the capital of New York
and is located on Long Island.
3 (2) 5 is not an odd number or 6 4 12.
4 (4) Daylight saving time ends in November
or the clock is not turned back one hour.
5 False
2-2 Compound Statments
3
True
True
True
True
Answers will vary. One possible answer is:
Every square has four sides or every triangle
has three sides. True
11 Answers will vary. One possible answer is:
The sum of the measures of the angles of
every triangle is 180 and every triangle has
four angles. False
6
7
8
9
10
2-3 Conditionals
(pages 26–27)
1 (2) You take this medicine.
2 (1) I will stay in this afternoon.
3 (1) true
4 (1) true
5 (1) true
6 Conditional
7 Conjunction
8 Conditional
9 Disjunction
10 Negation
11 Answers will vary. x 18 is one possible
answer.
12 x 2
13 Any positive number or zero
14 Any perfect square that is not even
2 is one possible
15 Answers will vary. x √
answer.
16 True
17 False
18
The sun is It is not If the sun is
shining.
Sunny
Sun
showers
Cloudy
Stormy
4
raining. shining, it is
not raining.
T
T
T
T
F
F
F
F
T
F
T
T
Chapter 2: Logic
2-4 Converses, Inverses,
and Contrapositives
(pages 28–29)
1 (1) If you stop your car, then the traffic light
is red.
2 (3) If the sum of two numbers is not even,
then the numbers are not both even.
3 (1) If the merrygoround was not oiled, then
it will squeak.
4 (2) If we are on vacation, it is summer.
5 (4) If a triangle is not equilateral, then it
does not have three congruent angles.
6 (2) If I didn’t turn on the air conditioner,
then the house isn’t cool.
7 (2) disjunction, p ∨ q
8 Converse: If school is open, then it is September.
Inverse: If it is not September, then school is
not open.
Contrapositive: If school is not open, then it is
not September.
9 Converse: If the figure has four sides, then it
is a square.
Inverse: If the figure is not a square, then it
does not have four sides.
Contrapositive: If the figure does not have
four sides, then it is not a square.
10 Converse: If someone will trade desserts, then
I have cookies with my lunch.
Inverse: If I do not have cookies with my
lunch, then someone will not trade desserts.
Contrapositive: If someone will not trade
desserts, then I do not have cookies with
my lunch.
11 Converse: If my teacher will be able to read
my paper, then I typed it.
Inverse: If I do not type my paper, then my
teacher will not be able to read it.
Contrapositive: If my teacher will not be able
to read my paper, then I did not type it.
12 Converse: If I am late to school, then I didn’t
set my alarm.
Inverse: If I set my alarm, then I will not be
late to school.
Contrapositive: If I am not late to school, then
I set my alarm.
13 Converse: If I am not in the starting lineup,
then I will not go to practice.
Inverse: If I to go to practice, I will be in the
starting lineup.
Contrapositive: If I am in the starting lineup,
then I will go to practice.
14 Converse: If I am unhappy, then I did not
make honor roll.
Inverse: If I make honor roll, then I am happy.
Contrapositive: If I am happy, then I made
honor roll.
15 Converse: If I do not have money for the
movies, then I will go shopping.
Inverse: If I do not go shopping, then I will
have money for the movies.
Contrapositive: If I have money for the
movies, then I will not go shopping.
16 Converse: If I do not take a school bus, then I
live close to school.
Inverse: If I do not live close to school, then I
take a school bus.
Contrapositive: If I take a school bus, then I do
not live close to school.
17 Converse: If I don’t study Latin, then I will
study French.
Inverse: If I don’t study French, then I will
study Latin.
Contrapositive: If I study Latin, then I will not
study French.
18 a If the segments are congruent, then their
measures are equal. (True)
b If the measures of the segments are equal,
then the segments are congruent. (True)
2-5 Biconditionals
(pages 31–32)
1 (1) true for any value of x
2 (2) false for any value of x
3 (2) false
4 (1) If I am sleepy, then I did not get eight
hours of sleep and if I did not get eight hours
of sleep, then I am sleepy.
5 (3) Paul does not buy chicken or he does not
roast it.
6 (4) the conjunction of a conditional and its
converse
7 If I have money, then I earned it.
8 If tomorrow is Friday, then today is Thursday. (True)
9 Converse: If a number is rational, then it is a
repeating decimal.
Biconditional: A number is rational if and only
if it is a repeating decimal.
The biconditional is false for integers and
terminating decimals.
10 If the sum of the measures of two angles is
not 45, then the two angles are not complementary.
Any pair of complementary angles makes
this statement and original false.
Any pair of angles with a sum of measures
not equal to 45 or 90 makes this statement
and the original true.
2-6 Laws of Logic
(page 36)
1 Margaret is a doctor. (Disjunctive Inference)
2 I will have straight teeth. (Detachment)
3 I did not save money. (Modus Tollens)
4 I helped my friend. (Modus Tollens)
5 It will not rain. (Detachment)
6 Joe is not a teacher. (Disjunctive Inference)
7 I will not eat dinner with Michael. (Modus
Tollens)
8 I will not play on Saturday. (Detachment)
9 If I practice the violin, my friend will be
jealous. (Chain Rule)
10 If I don’t help my brother, he cannot play
football. (Chain Rule)
2-7 Proof in Logic
(page 38)
1 1. p ∨ q
2. r → ~q
3. r
4. ~q
5. p
2 1. a → b
2. c ∨ a
3. ~c
4. a
5. b
Law of Detachment (2, 3)
Law of Disjunctive Inference (1, 4)
Law of Disjunctive Inference (2, 3)
Law of Detachment (1, 4)
2-7 Proof in Logic
5
3 1.
2.
3.
4.
5.
4 1.
2.
3.
4.
5.
6.
7.
5 1.
2.
3.
4.
5.
6.
7.
8.
9.
6 1.
2.
3.
4.
5.
6.
7.
7 1.
2.
3.
4.
5.
6.
7.
8.
9.
8 1.
2.
3.
4.
5.
6.
7.
e ∨ ~f
~f → g
~e
~f
g
a∨b
b→c
c→d
b→d
~d
~b
a
s∨t
s→r
r→q
s→q
~q
~s
t
t→v
v
d→e
d∨f
h → ~e
h
~e
~d
f
~f → g
~f ∨ j
g → ~h
j→k
h
~g
f
j
k
x→z
x→y
~z
~y
~x
x∨t
t
Law of Disjunctive Inference (1, 3)
Law of Detachment (2, 4)
Chain Rule (2, 3)
Law of Modus Tollens (4, 5)
Law of Disjunctive Inference (1, 6)
Chain Rule (2, 3)
Law of Modus Tollens (4, 5)
Law of Disjunctive Inference (1, 6)
Law of Detachment (8, 7)
Law of Detachment (3, 6)
Law of Modus Tollens (1, 5)
Law of Disjunctive Inference (2, 6)
Law of Modus Tollens (3, 5)
Law of Modus Tollens (1, 6)
Law of Disjunctive Inference (2, 7)
Law of Detachment (4, 8)
Law of Modus Tollens (1, 3)
Law of Modus Tollens (2, 4)
Law of Disjunctive Inference (5, 6)
Chapter Review (pages 38–40)
1 (1) true
2 (1) true
3 (2) false
6
Chapter 2: Logic
4 (2) false
5 (1) If I am late for school, I did not set my
alarm.
6 (2) If Marie is not bowling today, then today
is not Monday.
7 (1) The statement is always true but its converse cannot be determined.
8 Hypothesis: The triangle has a 90 angle.
Conclusion: The square of the longest side of
a triangle is equal to the sum of the squares
of the other sides.
9 Hypothesis: A and B are alternate interior
angles.
Conclusion: They are congruent.
10 False
11 True
12 True
13 False
14 False
15 Inverse: If the altitude does not bisect the
base, the triangle is not isosceles.
Converse: If the triangle is isosceles, the altitude bisects the base.
Contrapositive: If the triangle is not isosceles,
the altitude does not bisect the base.
Biconditional: The altitude bisects the base if
and only if it is isosceles.
16 True
17 False
18 p is true; q is false.
19 True
20 Converse: If a number is rational, then it is an
integer. (True)
Conditional statement is true.
Biconditional: A number is an integer if and
only if it is rational.
21 Converse: If a number is not prime, then a
number is a perfect square. (True)
Conditional statement is true.
Biconditional: A number is a perfect square if
and only if it is not prime.
22 Converse: If a parabola is tangent to the
x-axis, then its roots are equal. (True)
Conditional statement is true.
Biconditional: The roots of a quadratic equation are equal if and only if the parabola is
tangent to the x-axis.
23 Converse: If a number is real, then it is
rational. (False)
Conditional statement is true.
2 , then x √
32 . (True)
24 Converse: If x 4 √
Conditional statement is true.
32 if and only if
Biconditional: x √
√
x 4 2.
25 Converse: If a relation is a function, then it
passes the vertical line test. (True)
Conditional statement is true.
Biconditional: A relation passes the vertical
line test if and only if it is a function.
26 If I study, then I will not get poor grades.
27 If I get poor grades, then I did not study.
28 If I do not get poor grades, then I study.
29 If my baby cousin cries, then she gets a
bottle.
30 If the figure is a rectangle, then two adjacent
sides form a right angle.
31 If a triangle is equilateral, then it is equiangular.
32 I like math.
33 The class president is a girl. (Law of Disjunctive Inference)
34 My teacher will be happy. (Law of Detachment)
35 I do not go to the park. (Law of Modus
Tollens)
36 If I learn to knit, I will save money. (Chain
Rule)
37 I eat lunch. (Law of Modus Tollens)
38 I feel good. (Law of Detachment)
39 Rich likes lacrosse. (Law of Disjunctive
Inference)
40 Marlene does not babysit. (Law of Modus
Tollens)
41 Jack is using his computer. (Law of Modus
Tollens)
42 If I read novels when my teacher assigns
them, I watch less television. (Chain Rule)
43 If I like school, I won’t get a job after school.
(Chain Rule)
44 1.
2.
3.
4.
5.
a
p∨t
a → ~t
~t
p
45 1.
2.
3.
4.
x→y
~z ∨ x
z
x
5.
46 1.
2.
3.
y
c∨a
~c
a
4.
5.
47 1.
2.
3.
4.
5.
6.
7.
a→b
b
c → ~d
d
~c
~b → c
b
a ∨ ~b
a
Law of Modus Tollens (1, 2)
48 1.
2.
3.
4.
5.
6.
7.
~m ∨ n
~m → r
r → ~q
~m → ~q
q
m
n
Chain Rule (2, 3)
8. n → z
9. z
Law of Detachment (3, 1)
Law of Disjunctive Inference
(2, 4)
Law of Disjunctive Inference
(2, 3)
Law of Detachment (1, 4)
Law of Disjunctive Inference
(1, 2)
Law of Detachment (4, 3)
Law of Modus Tollens (4, 3)
Law of Disjunctive Inference
(6, 5)
Law of Modus Tollens (4, 5)
Law of Disjunctive Inference
(1, 6)
Law of Detachment (8, 7)
Chapter Review
7
CHAPTER
Introduction to
Geometric Proof
3
3-1 Inductive Reasoning
(page 42)
1 For a polygon with n sides and vertices, from
each vertex, diagonals can be drawn to n 3
other vertices. This can happen n times. Each
diagonal is counted twice from a to b and
n(n 3)
from b to a. There are _ diagonals.
2
Test:
Triangle
Quadrilateral
Pentagon
0
4(4 3)
_
2
2
5(5 3)
_
5
2
2 n5
12345
54321
−−−−−−−−−−−
6 6 6 6 6 5(6) n(n 1) twice
the sum
The sum of the integers from 1 through
n(n 1)
n _.
2
3 Answers will vary. 9, 15, 21 are some possible answers.
4 Answers will vary. 3, 9, 15, 21 are some possible answers.
3-2 Definitions and Logic
(page 43)
1 Line segments are congruent if and only if
they have the same measure.
2 A point of a line segment is the midpoint if
and only if it divides the line segment into
two congruent segments.
8
3 A line is an angle bisector if and only if
it divides an angle into two congruent
parts.
4 Lines are perpendicular if and only if the
lines form right angles.
5 A figure is a polygon if and only if it is a
closed figure in the plane formed by three
or more segments joined at their endpoints.
Chapter 3: Introduction to Geometric Proof
3-3 Deductive Reasoning
(page 44)
1 1. Given
2. Perpendicular lines form right angles.
3. Right triangles have one right angle.
divides FEH into two adjacent angles
2 EG
with the same measure. FEG and HEG
are congruent because they have the same
measure. An angle bisector is the ray that
divides an angle into two adjacent congruent
angles.
3 1. Given
2. Line segments equal in measure are
congruent.
3. A midpoint divides a line segment into
two congruent segments.
−−
4 Since PR bisects SPQ, it divides the angle
into two congruent angles. SPR QPR.
If two angles are congruent, they have the
same measure.
3-4 Indirect Proof
(page 45)
1 Statements
−−−
−−−
1. CD and HK are
not congruent.
2. CD HK
−−− −−−
3. CD HK
Reasons
1. Assumption.
2. Given.
3. Line segments
that are equal
in measure are
congruent.
Therefore, assumption is false.
Reasons
2 Statements
1. ABC is not a
right angle.
−− −−−
2. AB CD
3. ABC is a right
angle.
1. Assumption.
1. ABC is not an
isosceles triangle.
2. A B
3. ABC is an isosceles triangle.
1. Assumption.
2. Given.
3. Perpendicular
lines form right
angles.
Therefore, the assumption is false.
Reasons
3 Statements
2. Given.
3. An isosceles triangle contains two
congruent angles.
Therefore, the assumption is false.
Reasons
4 Statements
1. DB
is the angle bi- 1.
sector of ADC.
2.
2. mADB mBDC
3. DB
3.
is not the bisector of ADC.
Assumption.
Given.
An angle bisector
divides an angle
into two congruent parts.
Therefore, the assumption is false.
3-5 Postulates, Theorems,
and Proof
(pages 47–49)
1 Yes
2 No
3 No
4 The symmetric property of equality
5 The reflexive property of congruence
6 The symmetric property and transitive
property of congruence
Reasons
7 Statements
−− −−−
1. Given.
1. PQ QR
2. PQR is a right
angle.
2. Perpendicular
lines are two lines
that intersect to
form right angles.
3. mPQR 90
3. A right angle is
an angle whose
degree measure
is 90.
4. Given.
5. Perpendicular
lines are two
lines that intersect to form right
angles.
6. A right angle is
an angle whose
degree measure
is 90.
7. Symmetric property of equality.
8. Transitive property of equality.
−− −−
4. XY YZ
5. XYZ is a right
angle.
6. mXYZ 90
7. 90 mXYZ
8. mPQR mXYZ
8 Statements
Reasons
is the angle
1. AC
bisector of BAD.
1. Given.
2. BAC CAD
2. An angle bisector
divides an angle
into two congruent parts.
3. If two angles are
congruent, they
have the same
measure.
4. Given.
3. mBAC mCAD
is the angle
4. AD
bisector of CAE.
5. CAD DAE
5. An angle bisector
divides an angle
into two congruent parts.
3-5 Postulates, Theorems, and Proof
9
6. BAC DAE
7. mBAC mDAE
9 Statements
6. Transitive
property.
7. If two angles are
congruent, their
measures are
equal.
Reasons
1. 8 x y
1. Given.
2. y 3
2. Given.
3. 8 x 3
3. Substitution
property.
4. x 3 8
4. Symmetric
property.
Reasons
10 Statements
1. M is the midpoint 1. Given.
−−
of AB.
−−− −−−
2. A midpoint
2. AM MB
divides a line
segment into two
congruent line
segments.
−−− −−
3. Given.
3. MB BC
−−− −−
4. Transitive
4. AM BC
property.
3-6 Remaining Postulates
of Equality
(pages 51–52)
1 Partition postulate of equality
2 Division postulate of equality
3 Addition postulate
4 Subtraction postulate
5 Division postulate of equality
Note: Since there are many variations of proofs,
the following is simply one set of acceptable
statements to complete each proof. Depending
on the textbook used, the wording and format
of reasons may differ, so they have not been
supplied for the method of congruence applied
in each problem. (These solutions are intended
to be used as a guide—other possible solutions
may vary.)
10
Chapter 3: Introduction to Geometric Proof
6
1.
2.
3.
4.
5.
7 1.
2.
3.
4.
5.
6.
7.
8 1.
m1 m2
m3 m4
m1 m3 m2 m4
mDAB m1 m3
mBCD m2 m4
mDAB mBCD (Substitution
postulate)
−− −−
AB CB
−−− −−
AD CE
AB CB
AD CE
AB AD CB CE
DB EB
−− −−
DB EB
(Line segments
that are equal in
measure are congruent.)
AB AC
1 AC
2. AD _
3
1
_
3. AE AB
3
4. AD AE
(Division
postulate)
9 1. mEAB mFBC
is the angle bisector of EAB.
2. AG
3. BH
is the angle bisector of FBC.
1 mEAB
4. m1 _
2
1 mFBC
5. m2 _
2
6. m1 m2
(Division
postulate)
10 1. AB DE
2. AC 3AB
3. DF 3DE
4. AC DF
(Multiplication
postulate)
Chapter Review (page 52)
1 Reflexive property of equality
2 Transitive property of equality
3 Symmetric property of equality
1 mBAC
4 mBAD _
2
Note: Since there are many variations of proofs,
the following is simply one set of acceptable
statements to complete each proof. Depending
on the textbook used, the wording and format
of reasons may differ, so they have not been
supplied for the method of congruence applied
in each problem. (These solutions are intended
to be used as a guide—other possible solutions
may vary.)
−−−−
5 1. CMD is a line segment.
−−− −−−
2. CM MD
3. M is the mid- (Definition of a midpoint)
−−−
point of CD.
6 1.
2.
3.
4.
5.
6.
7.
7 1.
2.
3.
4.
5.
6.
7.
8 1.
2.
3.
4.
5.
−−−−
ABCD is a line segment.
−− −−−
AB CD
BC BC
AB CD
AB BC CD BC
AC BC
−− −−
AC BD
(Line segments that are
equal in measure are
congruent.)
−− −−−
AB CD
AB CD
−−−
−−
AB and CD bisect each other at E.
1 AB
AE _
2
1
_
CE CD
2
AE CE
−− −−
AE CE
(Line segments that are
equal in measure are
congruent.)
−− −−−
PQ QR
mPQR 90
mPQR mABC
mABC 90
−− −−
AB BC
(Perpendicular lines form
right angles.)
9 1.
2.
3.
4.
5.
6.
7.
10 1.
2.
3.
4.
5.
6.
11 1.
2.
4.
5.
6.
7.
8.
12 1.
2.
3.
4.
5.
6.
7.
8.
PQ XY
QR YZ
PQ QR XY YZ
PQ QR PR
XY YZ XZ
PR XZ
−− −−
PR XZ
(Line segments that are
equal in measure are
congruent.)
DAB and EBA are straight angles.
mDAB 180
mEBA 180
m1 m2
mDAB m1 mEBA m2
m3 m4
(Substitution postulate)
−− −−
AC BC
AC BC
−−− −−
DC EC
DC EC
AD DC BC EC
AD BE
−−− −−
AD BE
(Line segments that are
equal in measure are
congruent.)
−−− −−
AD BC
AD BC
−−−
E is the midpoint of AD.
1 AD
DE _
2
−−
F is the midpoint of BC.
1 BC
BF _
2
BF DE
−− −−
BF DE
(Line segments that are
equal in measure are
congruent.)
Chapter Review
11
CHAPTER
Congruence of Lines,
Angles, and Triangles
4
4-1 Setting Up a Valid
Proof
(pages 56–57)
, BD
1 Given: ABC
AC
Statements
1. PQRS
−− −−
2. PR QS
−−− −−−
3. QR QR
D
4. PR QR QS QR
−− −−
5. PQ RS
A
B
C
Prove: ABD CBD
Statements
1. ABC
2. BD
AC
3. ABD is a right
angle.
4. CBD is a right
angle.
5. ABD CBD
−− −−
, PR QS
2 Given: PQRS
P
Q
R
S
Reasons
1. Given.
2. Given.
3. Definition of perpendicular lines.
4. Definition of perpendicular lines.
5. Congruent angles
are angles that
have the same
measure.
−−− −−
3 Given: AD BC
−−−
−−
EF bisects AD
−−
−−
GF bisects BC
−− −−
Prove: AE BG
Statements
−−− −−
1. AD BC
−−−
−−
2. EF bisects AD
−−
−−
3. GF bisects BC
−− −−−
4. AE AD
−− −−
5. BG GC
6. AE ED
7. BG GC
−− −−
Prove: PQ RS
8. AE ED AD
12
Chapter 4: Congruence of Lines, Angles, and Triangles
Reasons
1. Given.
2. Given.
3. Reflexive
property of
congruence.
4. Subtraction
postulate
5. Substitution
postulate
Reasons
1. Given.
2. Given.
3. Given.
4. Definition of a
bisector.
5. Definition of a
bisector.
6. Definition of
congruent
segments.
7. Definition of
congruent
segments.
8. Partition
postulate.
9. BG GC BC
10. AE AE AD
or 2AE AD
9. Partition
postulate.
10. Substitution
postulate.
11. BG BG BC or 11. Substitution
postulate.
2BG BC
1
12. Division
12. AE _AD
2
postulate.
1
_
13. Division
13. BG BC
2
postulate.
14. Definition of
14. AD BC
congruent
segments.
1
_
15. Substitution
15. AE BC
2
postulate.
1
_
16. Symmetric
16. BC BG
2
property.
17. Transitive
17. AE BG
property.
−− −−
18. Segments with
18. AE BG
equal measures
are congruent.
4 Given: URS UTS
−−
US bisects RUT
−−
US bisects RST
Prove: 1 2
Statements
Reasons
1. Given.
1. URS UTS
−−
2. US bisects RUT 2. Given.
−−
3. US bisects RST
3. Given.
4. Definition of an
4. 1 TUS
angle bisector.
5. Definition of an
5. RSU 2
angle bisector.
6. Congruent
6. m1 mTUS
angles are equal
in measure.
7. Congruent
7. mRSU m2
angles are equal
in measure.
8. Partition
8. m1 mTUS
postulate.
mRUT
9. mRSU m2
mRST
9. Partition
postulate.
10. m1 m1 mRUT or
2(m1) mRUT
10. Substitution
postulate.
11. Substitution
11. m2 m2 postulate.
mRST or
2(m2) mRST
1 mRUT 12. Division
12. m1 _
2
postulate.
1
13. m2 _mRST 13. Division
2
postulate.
14. Congruent an14. mRUT gles are equal in
mRST
measure.
1
_
15. m1 mRST 15. Substitution
2
postulate.
1
_
16. Symmetric
16. mRST 2
2
property.
17. Transitive
17. m1 m2
property.
18. Congruent an18. 1 2
gles are equal in
measure.
−− −−
5 Given: HJ FK
−−− −−
Prove: HK FJ
Statements
−− −−
1. HJ FK
2. HJ FK
3. JK JK
4. HJ JK HK
5. FK JK FJ
6. HJ JK FK JK
7. HK FJ
−−− −−
8. HK FJ
Reasons
1. Given.
2. Definition of congruent segments.
3. Reflexive
property.
4. Partition
postulate.
5. Partition
postulate.
6. Addition
postulate.
7. Substitution
postulate.
8. Segments with
equal measures
are congruent.
4-1 Setting Up a Valid Proof
13
bisects CBE
6 Given: BA
1 2
2 4
Prove: 3 4
Narrative Proof: A bisector divides an angle
into two congruent angles, so 1 2.
Using the transitive property of congruence,
1 4. By substitution, 3 4.
7 4x 3 2x 21
x 12
mBCA 4x 3
4(12) 3
45
mBCE 2(45) 90
8 4x 4 2(3x 21)
x 23
AB 4x 4
4(23) 4
96
9 Since DE CE, then 2x y 5y, and
x 2y.
AC BD
6 5y (2x y) (x y)
6 5y 3x 2y
6 5y 3(2y) 2y
6 5y 6y 2y
6 3y
2y
x 2(2) 4
BD 2(4) 2 4 2 16
10 AB DE 8, so BD 4 and CD 2. Therefore, CE 8 2 10.
4-2 Proving Theorems
About Angles
(pages 59–61)
1 If two angles are right angles, they both have
a measurement of 90. Angles with the same
measure are congruent.
2 If two angles are straight angles, they both
have a measurement of 180. Angles with the
same measure are congruent.
3 If 1 is a complement of A, then m1 90 mA. If 2 is a complement of A,
then m2 90 mA. Then m1 m2
by the symmetric property and transition,
and 1 2 by definition of congruence.
14
4 If 1 2, 3 is the complement of
1, and 4 is the complement of 2, then
m3 90 m1 and m4 90 m2.
Since m1 m2 by congruence, m3 m4 by symmetric property and transition,
and 3 4 by congruence.
5 If 1 is a supplement of A, then m1 180 mA. If 2 is a supplement of A,
then m2 180 mA. Then m1 m2
by the symmetric property and transition,
and 1 2 by definition of congruence.
6 If 1 2 and 3 is the supplement of
1 and 4 is the supplement of 2, then
m3 90 m1 and m4 90 m2.
Since m1 m2 by congruence, m3 m4 by symmetric property and transition,
and 3 4 by congruence.
7 By definition the sum of their measures is a
straight angle or 180.
8 The two angles are a linear pair so the sum
of their measures is 180. If they are congruent, each measure is 90; they form right
angles and are therefore perpendicular.
9 Each angle forms a linear pair with the same
angle. They are supplementary to the same
angle and are therefore congruent.
10 m2 m3 because they are vertical angles. By substitution, 1 is complementary
to 3. 1 4 because complements of the
same angle are congruent.
11 3 forms a linear pair with 1, 3 is supplementary to 1. Since 1 2, 3 is
supplementary to 2. 4 forms a linear pair
with 2, 4 is supplementary to 2.
3 4 because supplements of the same
angle are congruent.
12 Since they form a linear pair, 2 is supplementary to EDG. Since 1 and 2 have
the same measure (they are congruent), 1 is
supplementary to EGD.
13 Since 1 is complementary to 3,
m1 m3 90. Since ACB is a right
angle, m1 m2 90. By subtraction and
substitution, m2 m3, so they are congruent angles.
14 By the transitive postulate, 1 3.
Because they are vertical angle pairs,
1 4, and 3 6. By substitution
and transition, 4 6.
Chapter 4: Congruence of Lines, Angles, and Triangles
15 x 25 180
x 155
16 2x x 180
x 60
17 x 85
18 (x 10) x 90
x 40
19 60 x x x 180
x 40
20 6x 20 4x 10
x 15
21 40 mz 180
mz 140
22 a mDOE 75
b mAOB 15
c mDOC 105
23 2x 15 4x 1
x7
mPRT 2(7) 15 29
mRTQ 180 mPRT
180 29
151
24 3x 25 10x 4
x3
mWVY 3(3) 25 34
mWVZ 180 mWVY
180 34
146
25 (5x 2y) (5x 2y) 180
x 18
mAEC mBED
11y 5x 2y
11y 5(18) 2y
9y 90
y 10
mAED mCEB 5x 2y
5(18) 2(10) 70
4-3 Congruent Polygons
(pages 64–66)
1 (3) IHG MNL
2 (4) a side of one is congruent to a side of the
other
3 (4) IV
4 (1) I
5 (2) II
−−
6 a AC
b BAC
−−
c BC
d CBA
7 a ACD BCD
b ADC CBA
8 a mE 44
b mB 110
c mC 26
9 Answers will vary. For instance, a 3-4-5
triangle has the same angle measure as a
6-8-10 triangle. They are not congruent.
10 Answers will vary.
11 Answers will vary.
12 a Vertical angles are congruent; the triangles
are congruent by SAS.
b AC BD
2x x 5
x5
AC BD 10
AE BE 11
CE DE 12
13 The triangles are congruent by SAS.
ACE BDE by corresponding parts of
congruent triangles are congruent. Therefore,
ACE BDE. So 5x 7x 18, and x 9.
mACE 45; mBDE 45; mBED 45.
mDBE 180 45 45 90. Therefore,
DBE is a right triangle.
2
14
x 4x x 14
2
x 5x 14 0
(x 7)(x 2) 0
x 7 and x 2
When x 7, AB AB 21
When x 2, AB AB 12
15 No, the angle must be included between the
pairs of sides.
(page 67)
Note: Since there are many variations of proofs,
the following is simply one set of acceptable
statements to complete each proof. Depending
on the textbook used, the wording and format
of reasons may differ, so they have not been
supplied for the method of congruence applied
in each problem. (These solutions are intended
to be used as a guide—other possible solutions
may vary.)
4-3 Congruent Polygons
15
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
2 1.
2.
3.
4.
5.
1
6.
1.
2.
3.
4.
5.
6.
7.
8.
4 1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
12.
5 1.
2.
3.
4.
5.
6.
6 1.
2.
3.
3
16
−−−−
ABCD
1
___ 2
−−
AF DE
−− −−
AC BD
AC BD
BC BC
(Reflexive property)
AC BC BD BC
AB CD
−− −−−
AB CD
ABF DCE
(SAS SAS)
−−− −−−
ABC, DEF
3 4
−− −−
BF EC
1 2
−−− −−−
FBC CEF
(If two angles are
congruent, their
supplements are
congruent.)
BCF EFL
(ASA ASA)
−−
−−
BE is a median to FD.
−− −−
FE DE
−− −−
BE BE
−−− −−
AD CF
−− −−
AB CB
AD AB CF CB
−− −−
BD BF
FBE DBE
(SSS SSS)
−− −−−
AE DC
−− −−
DE DE
AE DE DC DE
−−− −−
AD EC
3 4
ADB and 3 are linear pairs.
ADB and 3 are supplements.
CEB and 4 are linear pairs.
CEB and 4 are supplements.
ADB CEB
1 2
ADB CEB
(ASA ASA)
−−
D is the midpoint of AB.
AD DB
−−− −−
AD DB
−− −−
AC BC
−−− −−−
DC DC
ADC BDC (SSS SSS)
−− −−
AB BC
−− −−
EF AB
−− −−
EF BC
4.
5.
6.
7.
7 1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
12.
8 1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
−−
−−
BE bisects CF at D.
−−− −−
CD DF
1 2
BCD EFD
(SAS SAS)
−− −−
AC BD
−− −−
BC BC
AC BC BD BC
−− −−−
AB DC
3 4
3 and FBA are linear pairs.
3 and FBA are supplements.
4 and ECD are linear pairs.
4 and ECD are supplements.
FBA ECD
1 2
EDC FAB
(ASA ASA)
−−
is the perpendicular bisector of AB.
EG
−− −−
AF BF
EFA is a right angle.
EFB is a right angle.
EFA EFB
mEFA m1 mEFB m1
1 2
mEFA m1 mEFB m2
DFA CFB
−− −−
DF CF
ADF BCF
(SAS SAS)
Chapter Review (pages 68–69)
1 Given: ABC and DBE are vertical angles.
PQR ABC
Prove: PQR DBE
intersect at E.
and CD
2 Given: AB
AEC FGH
Prove: CEB is supplementary to FGH.
are perpendicular lines.
and CED
3 Given: AEB
Prove: AEC AED
are perpendicular lines.
and CED
4 Given: AEB
.
F is not on CD
.
Prove: FE
is not perpendicular to AB
−−
is a bisec5 Since E is the midpoint of AB, CD
−−
tor of AB. Since AEC BEC and they are
a linear pair, the measure of each must be
180
_
90.
2
Chapter 4: Congruence of Lines, Angles, and Triangles
6 1 3 because they are vertical angles.
4 2 because they are vertical angles. By
the substitution postulate, 2 3. By the
transitive postulate, 4 3, or 3 4
by the symmetric property.
−−
7 Since PQ bisects ABC, then 2 CBQ.
Since CBQ and 1 are vertical pairs,
CBQ 1. By the transitive postulate,
2 1, or 1 2 by the symmetric
property.
8 Since CDE is a right triangle, the two angles that are not the right angle are complementary because there sum is 180 (whole
triangle) 90 (right angle) 90. Therefore,
FED is complementary to FCD. By substitution, EDF is complementary to FCD.
9 mAED mBED 180
4x 20 9x 48 180
x 16
mCEB mAED
4x 20
4(16) 20
84
10 mAED mCEB
5x 7x 26
x 13
mAED 5(13) 65
mAED mDEB 180
65 mDEB 180
mDEB 115
11 mAED mCEB
12x 6x 10y
6x 10y
mAED mDEB 180
12x 6x 180
18x 180
x 10
6x 10y
60 10y
y6
mAEC mDEB
10y
10(6) 60
12 mCEB mDEB 180
9y 12y 12 180
y8
mAEC mDEB
12(8) 12 108
Note: Since there are many variations of proofs,
the following is simply one set of acceptable
statements to complete each proof. Depending
on the textbook used, the wording and format
of reasons may differ, so they have not been
supplied for the method of congruence applied
in each problem. (These solutions are intended
to be used as a guide—other possible solutions
may vary.)
−− −−
13 1. AC DF
2. A D
3. C F
4. ABC DEF
(ASA ASA)
−− −−
14 1. HE FE
−−− −−
2. HG FE
−− −−
3. HF HF
4. EFH GHF
(SSS SSS)
15 1. A, B, C, D lie on circle O.
−−−− −−−−
2. AOC; BOD
−−− −− −−− −−−
3. AO, BO, CO, DO are radii.
−−− −−−
4. AD CO
−− −−−
5. BO DO
6. AOB COD
(If two lines
intersect, the
vertical angles are
congruent.)
7. ABO CDO
(SAS SAS)
−− −−
16 1. QT RT
2. QT RT
−− −−
3. TV TU
4. TV TU
5. QT TV RT TV
6. QT TV RT TU
−−− −−−
7. QTV; RTU
8. QU RU
−−− −−−
9. QV RU
−− −−
10. PU VS
−−
−−− −− −−−
11. PU UV VS UV
−− −−
12. PV SU
−− −−
13. PQ RS
14. PQU SRU
(SSS SSS)
−−
17 1. C is the midpoint of AE.
−− −−
2. AC CE
3. 1 2
4. 3 BCD
5. 4 BCD
6. 3 4
7. ABC EDC
(ASA ASA)
Chapter Review
17
18
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
12.
−−
I is the midpoint of EG.
−− −−
EI IG
EI IG
−− −− −−
EG EI IG
EG EI IG
EG EI EI or EG 2EI
EH 2EI
EG EH
−− −−
EG EH
FGE IEH
FEG HIE
FGE IEH
(ASA ASA)
19 1.
2.
3.
4.
5.
20 1.
2.
3.
4.
5.
6.
7.
8.
9.
AEB, CED are right angles.
AEB CED
A C
−− −−
AE CE
ABC CDE
(ASA ASA)
−−− −−−
DC AD
−− −−−
AB AD
−−− −−
DC AB
−− −−
DF BE
−− −−
EF EF
DF EF BE EF
−− −−
DE BF
1 2
ABF DCE
(SAS SAS)
CHAPTER
Congruence Based on
Triangles
5
5-1 Line Segments
Associated With Triangles
(page 71)
1 One line
2
C
Median
A
3
4
5
6
18
Altitude
E F D
Angle Bisector
B
−−
−−
AE and BE
−−−
−−
AB and CD
ACF and BCF
ADC and BDC
Chapter 5: Congruence Based on Triangles
5-2 Using Congruent
Triangles to Prove Line
Segments Congruent and
Angles Congruent
(pages 72–73)
Note: Since there are many variations of proofs,
the following is simply one set of acceptable
statements to complete each proof. Depending
on the textbook used, the wording and format
of reasons may differ, so they have not been
supplied for the method of congruence applied
in each problem. (These solutions are intended
to be used as a guide—other possible solutions
may vary.)
1 1.
2.
3.
4.
5.
−− −−−
AB CD
−− −−−
BC DA
−− −−
AC AC
ABC CDA
BAC DCA
2 1.
2.
3.
4.
5.
−− −−
BA BC
−−− −−−
DA DC
−− −−
DB DB
ABD CBD
ABC CBD
3 1.
2.
3.
4.
5.
1 3
2 4
−− −−
AC AC
DAC BCA
−−− −−
AD CB
4
5
6
7
(SSS SSS)
(Corresponding
parts of congruent
triangles are
congruent.)
(SSS SSS)
(Corresponding
parts of congruent
triangles are
congruent.)
(ASA ASA)
(Corresponding
parts of congruent
triangles are
congruent.)
−−−
1. CD is the median drawn from C.
−−− −−
2. AD DB
−−− −−
3. CD AB
4. ADC is a right angle.
5. BCD is a right angle.
6. ADC BDC
−−− −−−
7. CD CD
8. ADC BDC
(SAS SAS)
−− −−
(Corresponding
9. CA CB
parts of congruent
triangles are
congruent.)
RS RS
4x 1 3x 3
x4
RT RT x 6 4 6 10
AD CB
2x 5 3x 7
x 12
AB x 10 12 10 22
CD AB 22
AD 2x 5 2(12) 5 29
CB 3x 7 3(12) 7 29
The triangle is isosceles and RS RT.
6x 4 3x 11
x5
Note: Since there are many variations of proofs,
the following is simply one set of acceptable
statements to complete each proof. Depending
on the textbook used, the wording and format
of reasons may differ, so they have not been
supplied for the method of congruence applied
in each problem. (These solutions are intended
to be used as a guide—other possible solutions
may vary.)
8 1. 1 3
2. 2 4
−− −−−
3. AB CD
4. ABE CDE
(ASA ASA)
−− −−
5. AE CE
−−
−−
6. BD bisects AC.
−− −−
7. BE DE
−−
−−
(Definition of a
8. AC bisects BD.
bisector)
−−−
9 1. DA bisects BDF.
2. FDA BDA
3. 1 2
4. m1 mFDA m2 mFDA
5. m1 mFDA m2 mBDA
6. EDA CDA
−−− −−
7. CD DE
−−− −−−
8. AD AD
9. EDA CDA
(SAS SAS)
−− −−
(Corresponding
10. AE AC
parts of congruent
triangles are
congruent.)
−−
10 1. BA is a median of CBF.
−− −−
2. CA FA
−−− −−
3. CD FE
−− −−−
4. FC CD
−− −−
5. FC EF
6. DCA is a right angle.
7. EFA is a right angle.
8. DCA EFA
9. DCA EFA
(SAS SAS)
−−− −−
(Corresponding
10. DA EA
parts of congruent
triangles are
congruent.)
5-2 Using Congruent Triangles to Prove Line Segments Congruent and Angles Congruent
19
5-3 Isosceles and
Equilateral Triangles
6. SRU TRU
(page 74)
−−−
7. RU bisects SRT.
1 Statements
1. Construct a
median from the
vertex formed
by the two congruent sides of
the triangle, to
form RST with
−−−
median RU.
−− −−
2. RS RT
−− −−
3. SU TU
−−− −−−
4. RU RU
5. RSU RTU
6. RSU RTU
2 Statements
1. Construct a
median from the
vertex formed
by the two congruent sides of
the triangle, to
form RST with
−−−
median RU.
−− −−
2. RS RT
−− −−
3. SU TU
−−− −−−
4. RU RU
5. RSU RTU
20
Reasons
1. A median of a
triangle is a line
segment with one
endpoint at any
vertex of the
triangle, extending to the
midpoint of the
opposite side.
2. Definition of an
isosceles triangle.
3. Definition of a
median.
4. Reflexive
property of
congruence.
5. SSS SSS.
6. Corresponding
parts of congruent triangles are
congruent.
Reasons
1. A median of a
triangle is a line
segment with one
endpoint at any
vertex of the triangle, extending
to the midpoint of
the opposite side.
2. Definition of an
isosceles triangle.
3. Definition of a
median.
4. Reflexive property
of congruence.
5. SSS SSS.
Chapter 5: Congruence Based on Triangles
3 Statements
1. Construct a
median from the
vertex formed
by the two congruent sides of
the triangle, to
form RST with
−−−
median RU.
−− −−
2. RS RT
−− −−
3. SU TU
−−− −−−
4. RU RU
5. RSU RTU
6. SUR TUR
7. SUR is a right
angle; TUR is a
right angle.
−−− −−
8. RU ST
4 Statements
1. Construct a
median from the
vertex formed
by the two congruent sides of
the triangle, to
form ABC with
−−−
median AX.
−− −−
2. AB AC
6. Corresponding
parts of congruent triangles are
congruent.
7. Definition of
angle bisector.
Reasons
1. A median of a
triangle is a line
segment with one
endpoint at any
vertex of the
triangle, extending to the
midpoint of the
opposite side.
2. Definition of an
isosceles triangle.
3. Definition of a
median.
4. Reflexive property
of congruence.
5. SSS SSS.
6. Corresponding
parts of congruent triangles are
congruent.
7. Adjacent congruent angles are
supplementary.
8. Definition of perpendicular lines.
Reasons
1. A median of a
triangle is a line
segment with
one endpoint at
any vertex of
the triangle,
extending to the
midpoint of the
opposite side.
2. Definition of
an equilateral
triangle.
−− −−
3. BX XC
−−− −−−
4. AX AX
5. ABX ACX
6. ABX ACX
7. Construct a sec−−
ond median, BY,
from vertex B.
−− −−
8. BA BC
−− −−
9. AY YC
−− −−
10. BY BY
11. BAY BCY
12. BAY BCY
13. ABC is equiangular.
3. Definition of a
median.
4. Reflexive
property of
congruence.
5. SSS SSS.
6. Corresponding
parts of congruent triangles are
congruent.
7. A median of a
triangle is a line
segment with
one endpoint
at any vertex
of the triangle,
extending to the
midpoint of the
opposite side.
8. Definition of an
equilateral
triangle.
9. Definition of a
median.
10. Reflexive
property of
congruence.
11. SSS SSS.
12. Corresponding
angles of congruent triangles are
congruent.
13. BCY is the
same as ACX,
and all three
angles of the triangle are equal.
5 mA mC 3x 5
mA mB mC 180
(3x 5) (4x 10) (3x 5) 180
10x 180
x 18
mC 3(18) 5 59
6 The measure of each angle is 60. So,
3x 6 60; x 22. And 3y 6 60;
y 18.
7 Show that RXZ TYZ SYX by
−− −− −−
SAS SAS, and XY YZ ZX because
corresponding parts of congruent triangles
are congruent.
−−
−−
8 Construct AC and BD, and label their inter−−
−−
section E. ABE CBE. AC and BD are
perpendicular. AED and CDE are right
angles; therefore, AED CDE.
−− −− −− −−
AE CE. ED ED. AED CED.
CAD ACD.
9
mA mB mC 180
(6x 12) (8x 8) (3x 6) 180
17x 10 180
x 10
mA 72; mB 72; mC 36
The triangle is isosceles.
10 mB 180 2x
5-4 Working With Two
Pairs of Congruent
Triangles
(pages 75–76)
Note: Since there are many variations of proofs,
the following is simply one set of acceptable
statements to complete each proof. Depending
on the textbook used, the wording and format
of reasons may differ, so they have not been
supplied for the method of congruence applied
in each problem. (These solutions are intended
to be used as a guide—other possible solutions
may vary.)
1 1. ABC EFG
−− −−
2. AC EG
−−
3. BD is a median.
4. D is the midpoint
(Definition of
−−
median)
of AC.
1
5. DC _AC
(Definition of mid2
point)
−−
6. FH is a median.
−−
7. H is the midpoint of EG.
1 EG
8. HG _
2
9. AC EG
1 AC
10. HG _
2
11. HG DC
5-4 Working With Two Pairs of Congruent Triangles
21
−−− −−−
12. HG DC
−− −−
13. FG BC
14. BCD FGH
15. BCD FGH
(SAS SAS)
−− −−
16. BD FH
−−
−−
2 a 1. PR and QS bisect each other at V.
−− −−
2. PV RV
−−− −−
3. QV SV
4. PVQ RVS
(Vertical angles are
congruent.)
5. PQV RSV
(SAS SAS)
b A continuation of part a.
6. UPV TRV
(Corresponding
parts of congruent
triangles are
congruent.)
7. PVU RVT
(Vertical angles are
congruent.)
8. PUV RTV
(ASA ASA)
−− −−−
3 a 1. AC AD
−− −−
2. BC BD
−− −−
3. AB AB
4. ABC ABD
(SSS SSS)
5. BAC BAD
(Corresponding
parts of congruent
triangles are
congruent.)
−− −−
6. AE AE
7. ACD ADE
(SAS SAS)
b A continuation of part a.
8. 1 2
(Corresponding
parts of congruent
triangles are
congruent.)
−−− −−
4 1. CD CB
2. 1 2
−− −−
3. CF CF
4. CFB CFD
(SAS SAS)
5. CBF CDF
(Corresponding
parts of congruent
triangles are
congruent.)
6. EDF ABF
7. BFA EFD
−− −−
8. DF BF
9. BFA DFE
(ASA ASA)
10. 3 4
(Corresponding
parts of congruent
triangles are
congruent.)
22
Chapter 5: Congruence Based on Triangles
5 1.
2.
3.
4.
5.
6.
7.
8.
−− −−
AC BC
−− −−
AE BE
−− −−
EC EC
ACE BCE
ACE BCE
−−− −−−
CD CD
ACD BCD
−−− −−
AD BD
(SSS SSS)
(SAS SAS)
(Corresponding
parts of congruent
triangles are
congruent.)
−− −−−
SP QR
1 2
−−
−−
AB bisects QS at C.
−− −−−
SC QC
(Definition of
bisector)
−− −−
5. SQ SQ
6. SQP QSR
(SAS SAS)
7. SQP QSR
(Corresponding
parts of congruent
triangles are
congruent.)
8. ACQ BCS
9. ACQ BCS (ASA ASA)
b Continuation of part a.
−− −−
(Corresponding
10. AC BC
parts of congruent
triangles are
congruent.)
6 a
1.
2.
3.
4.
5-5 Proving Overlapping
Triangles Congruent
(page 77)
Note: Since there are many variations of proofs,
the following is simply one set of acceptable
statements to complete each proof. Depending
on the textbook used, the wording and format
of reasons may differ, so they have not been
supplied for the method of congruence applied
in each problem. (These solutions are intended
to be used as a guide—other possible solutions
may vary.)
−− −−
1 1. AE BE
2. EAB EBA (If two sides of
a triangle are congruent, the angles
opposite these sides
are congruent.)
3.
4.
5.
2 1.
2.
3.
4.
5.
6.
7.
3 1.
2.
3.
4.
5.
4 1.
2.
3.
4.
5.
6.
7.
8.
9.
5 1.
2.
3.
4.
5.
6 1.
2.
3.
4.
5.
6.
7.
8.
−− −−
AB AB
−− −−
AC BD
ABC BAD
(SAS SAS)
1 3
2 2
1 3 2 3
ADB EDC
(Partition
postulate)
4 5
−−− −−
AD ED
ADB EDC
(ASA ASA)
−− −−
AE BE
−−
D is the midpoint of AB.
−−− −−
AD BD
−− −−
ED ED
ADE BDE
−− −−
TP TQ
TPQ TQP
SPQ RQP
SPT RQT
STP RTQ
(Vertical angles are
congruent.)
STP RTQ
(SAA SAA)
−− −−
SP PQ
−− −−
PQ PQ
SPQ RQP
−− −−
EF GF
−− −−
DF HF
DFH DFH
DFG EFH
(SAS SAS)
1 2
(Corresponding
parts of congruent
triangles are
congruent.)
−−
−−−
AD and FC bisect each other at G.
−−− −−−
AG DG
−− −−
FG CG
AGC DGF
(Vertical angles are
congruent.)
AGC DGF
(SAS SAS)
GDE GAC
(Corresponding
parts of congruent
triangles are
congruent.)
AGB DGE
AGB DGE
(ASA ASA)
5-6 Perpendicular
Bisectors of a Line
Segment
(pages 79–80)
−−
1 Given AB and P and Q, such that PA PB
−−
−−
and QA QB, let AB intersect PQ at
point M. APQ BPQ. APM BPM.
−−− −−−
APM BPM. AM BM. Therefore,
AMP BMP.
−−
2 a) Given AB and point P such that PA PB;
if two points are each equidistant from the
endpoints of a line segment, then the points
determine the perpendicular bisector of the
line segment. Select a second point that is
equidistant from both A and B, for example
−−
−−−
the midpoint of AB, M. Then PM is the per−−
pendicular bisector of AB.
b) Given point P on the perpendicular bisec−−
−−
tor of AB, let point M be the midpoint of AB.
PMA PMB.
−−
3 The perpendicular bisector of PQ
Note: Since there are many variations of proofs,
the following is simply one set of acceptable
statements to complete each proof. Depending
on the textbook used, the wording and format
of reasons may differ, so they have not been
supplied for the method of congruence applied
in each problem. (These solutions are intended
to be used as a guide—other possible solutions
may vary.)
−− −−
4 1. JL KL
−− −−−
2. JM KM
3. JL KL
(Congruent segments are
equal in length.)
4. JM KM
−−−
−−
5. LM is the perpendicular bisector of JK.
(Two points, each equidistant from the
endpoints of a line segment, determine
the perpendicular bisector of the line
segment.)
6. N is the midpoint of JK.
(A point on the perpendicular bisector
of a line segment is equidistant from the
endpoints of the line segment.)
−− −−−
(Definition of a midpoint)
7. JN KN
5-6 Perpendicular Bisectors of a Line Segment
23
−−
−−
5 1. FG is the perpendicular bisector of HI.
−−
−−
2. JG is the perpendicular bisector of HI.
(F, J, and G are collinear.)
−− −
3. HJ IJ
−− −−
4. GJ GJ
5. HJG and IJG are right angles.
(Definition of perpendicular bisector)
6. HJG IJG
(Right angles are
congruent.)
7. HGJ IGJ
(SAS SAS)
8. 1 2 (Corresponding parts of
congruent triangles are congruent.)
−−−
−−
6 1. AB and CD bisect each other.
−− −−−
2. AB CD
−− −−−
3. AC AD
−− −−
4. BC BD
−−− −−
5. AD BD
−− −−
6. AC BC
7. ACBD is an equilateral quadrilateral.
(Definition of equilateral quadrilateral)
7 1. 1 2
2. 3 4
−− −−
3. XZ XZ
4. XVZ XWZ (ASA ASA)
−−− −−−
(Corresponding parts
5. XV XW
of congruent triangles
are congruent.)
6. XV XW
−−− −−−−
7. XZY; VYW
−−
−−−−
8. XY is the perpendicular bisector of VYW.
(A point equidistant from the endpoints
of a line segment is on the perpendicular
bisector of the line segment.)
8
BP CP
4y 4
y1
AP CP
xy4
x14
x5
9
AP BP
3x y x y
xy
BP CP
xy4
xx4
x2
y2
10 CG AG 10
24
Chapter 5: Congruence Based on Triangles
5-7 Constructions
(page 83)
1 (4) BAC and GHI
2
A
B
3
C
V
T
R
S
4
A
B
C
D
E
F
5
1
2
3
6 If the legs of the compass form two sides of
a triangle, then the line segment connecting
the pencil to the point is the third side. If
the angle between the legs does not change
(and the legs remain the same length), then
any line segment connecting the legs is
congruent to the original because of SAS
congruence.
7
A
F
G
B
H
C
E
D
L
The first arc drawn in the construction creates an isosceles triangle, BGH. The rest of
the construction copies the sides of BGH to
a new triangle. ED BH, FD GH, and
FE GB, since F and G lie on the intersections. Therefore, BGH EFD by SSS
congruence and all corresponding angle
measures, including B and E, are equal.
8 Draw a point. Draw an arc centering on the
point. Draw two lines from the point intersecting the arc. Connect the points of intersection. This will form an isosceles triangle.
9
13
C
E
D
A
10
P
C
D
B
A
B
A
14
C
D
B
C
D
T
A
11
B
15
B
B
M
D
C
A
12
A
C
D
B
16
A
A
C
D
B
C
5-7 Constructions
25
Chapter Review (pages 84–85)
Note: Since there are many variations of proofs,
the following is simply one set of acceptable statements to complete each proof. Depending on the
textbook used, the wording and format of reasons
may differ, so they have not been supplied for the
method of congruence applied in each problem.
(These solutions are intended to be used as a
guide—other possible solutions may vary.)
−−−
1 1. MN bisects PNQ.
2. RND RNQ
3. RNP RQN
−−− −−−
4. RN RN
5. RPN RQN
(ASA ASA)
−− −−
2 1. PQ PR
−−
2. QT is a median.
−− −−
3. PT RT
−−
4. RS is a median.
−− −−
5. PS QS
6. PQT PRS
(SSS SSS)
3 1. 3 4
−− −−
2. DE DF
−−− −−−
3. DC DC
4. DEC DFC
(SAS SAS)
5. DCE DCF
(Corresponding
parts of congruent
triangles are
congruent.)
6. EDC FDC
7. mCAD m1 mEDC mDCE
180
8. mCBD m2 mFDC mDCF
180
9. mCAD mCBD
10. CAD CBD
11. ABC is isosceles. (Definition of isosceles triangle)
4 1. ABC
is
an
equilateral
triangle.
___
___
(Definition of equi2. AB AC
lateral triangle)
3. DCB DBC
4. DB
___ DC
___
5. AD AD
6. ABD ACD
(SSS SSS)
7. BAD CAD
(Corresponding
parts of congruent
triangles are
congruent.)
−−−
8. AD bisects BAC. (Definition of an
angle bisector)
26
Chapter 5: Congruence Based on Triangles
5
____
___
1. TM TA
2. MTA is isosceles.
___
(Definition of isosceles triangle)
___
___
(In an isosceles
triangle, the
bisector is the
altitude.)
___
(SSS SSS)
(Corresponding
parts of congruent
triangles are
congruent)
3. TH
___ bisects MTA.
4. TH is an altitude
of MTA.
6
1.
2.
3.
4.
5.
6.
7.
8.
9.
AB
AD
___ ___
CB
CD
___
___
AC AC
ABC ADC
BAE DAE
___
AE AE
ABE
ADE
___
___
BE DE
DAB is isoceles.
___
10. AE is the median
of DAB.
___
11. AE is the altitude
of DAB.
7
8
9
___
___
(ASA ASA)
(Definition of
isosceles triangle)
(Definition of
median)
(In an isosceles triangle, the median
is the altitude.)
AB BD
A D
DBA CBE
EBD EBD
DBA EBD CBE EBD
EBA CBD
ABE DBC
(ASA ASA)
ADE BDC
EDB EDB
ADE EDB BDC EDB
ADB EDC
DAE
DEC
___
___
DA DE
DAB DEC
(ASA ASA)
1 2
3
___ 4
___
DE DF
DEG____
DFH
(ASA ASA)
___
GD HD
(Corresponding
parts of congruent
triangles are
congruent.)
6. EDF EDF
1.
2.
3.
4.
5.
6.
7.
1.
2.
3.
4.
5.
6.
7.
1.
2.
3.
4.
5.
1 EDF 2 EDF
GDF EDH
FGD EDH
(ASA ASA)
5 6
(Corresponding
parts of congruent
triangles are
congruent.)
___
___
and
BD
bisect
each
other at G.
10 1. AC
___
____
AG
2. CG
___ ___
3. BG DG
4. AGB CGD
(Vertical angles are
congruent.)
5. AGB
CGD
(ASA ASA)
___
___
6. AB CD
7. GCD GAB
(Corresponding
parts of congruent
triangles are
congruent.)
8. 1 2
9. CED
AFB
(ASA ASA)
___
___
(Corresponding
10. EC FA
parts of congruent
triangles are
congruent.)
11 Assume that ___
ABC and
_____ABC are congruent and that BD and BD are angle bisectors
of B and B, respectively.
____
___
AB _____
AB. A A.
ABD ABD. ___
ABD ABC. BD BD.
12 1. ___
BIG___
is equilateral.
___
2. IA BC GT
3. __
IA ___
BC ___
GT
4. IB BG GI
5. IB BG GI
6. IB IA AB; BG BC CG;
GI GT TI
7. IA AB BC CG GT TI
8. IA AB IA BC CG BC GT TI GT
9. AB
TI
___ __
___ CG
10. AB CG TI
11. BIG IGB GBI
12. IAT BCA
(ASA ASA)
GTC
___
___
___
(Corresponding
13. AT CA TC
parts of congruent
triangles are
congruent.)
14. CAT is
(Definition of an
equilateral.
equilateral triangle)
7.
8.
9.
10.
15
16
17
18
19
___
RU
___
___
RT US
RT US
RT ST US ST
RS
___ TU
___
RS TU
R U
VST WTS
mVST mWTS
VSR is the complement of VST.
WTU is the complement of WTS.
VSR WTU
RVS UWT
(ASA ASA)
____
____
MQ
NQ
___
____
QP
QO
___
____
PQ MQ
MQP____
is a right angle.
____
OQ NQ
NQO is a right angle.
MQP NQO
(Right angles are
congruent.)
8. PQO PQO
9. MQP PQO NQO PQO
10. MQO NQP
11. MQO
NQP
(SAS SAS)
___
___
1. AC BC
2. ACF BCG
3. DCF ECG
4. DCF ACF BCG ACF
5. DCF ACF BCG ECG
6. ACD BCE
7. CAF CBA
8. ___
CAD___
CBE
(ASA ASA)
9. DC EC
10. DCE is isosceles. (Definition of an
isosceles triangle)
Draw a line longer than the sum of ___
the
lengths of the two segments. Copy AB onto
the new line. Place the compass vertex where
the arc swing___
intersects the line and mark off
the length of CD. The line segment from the
original vertex to the final arc swing marks
off the___
new segment.
Bisect AB and then bisect each half of the
original segment.
Use angle bisector
procedure.
___
Bisect side AB. Mark the point where the
bisector intersects the line M. Draw a line
from C to M.
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
12.
13.
14 1.
2.
3.
4.
5.
6.
7.
13
Chapter Review
27
20 Use constructing congruent angles procedure.
21 Use angle bisector procedure on each side of
the triangle. The angle bisectors should meet
at a common point, P.
22 Use constructing a perpendicular bisector
procedure.
23 Copy one side and the angles at each vertex.
Extend the rays until they meet.
24 Use the line bisector procedure. Mark the
intersection of the line and its bisector as the
midpoint.
CHAPTER
Transformations and
the Coordinate Plane
6-1 Cartesian Coordinate
System
(pages 91–92)
1 (2) 5
2 (2) 6
3 (4) 10
4 (2) 6
2
5 (1) √
√
226
6 (2) 7 a A(5, 2), B(3, 3) b A(3, 4), B(0, 5)
c A(x, y), B(0, 0) ___
8 The length of base AB is the difference
between the x-coordinates of A and B;
6 2 4.___
The height is the vertical distance
from C to AB or the difference between the
y-coordinates; 4 1 3. Therefore,
1 (4)(3) 6.
1 bh _
A_
___
2
2
9 The length of the base AB is the difference
between the x-coordinates of A and B;
1 (3) 4. The
___height is the vertical distance from C to AB or the difference between
the y-coordinates; 8 4 4. Therefore,
1 (4)(4) 8.
1 bh _
A_
2
2
28
6
−−
10 The length of the base AB is the difference in
the y-coordinates of A and B; 6 2 4. The
height is the horizontal distance from C
−−
to AB or the difference between the
x-coordinates; 3 2 1. Therefore,
1 (4)(1) 2.
1 bh _
A_
___
2
2
11 The length of the base AB is the difference
between the y-coordinates of A and B;
8 2 6. The height
is the horizontal dis___
tance from C to AB or the difference between
the x-coordinates; 4 (2) 6. Therefore,
1 (6)(6) 18.
1 bh _
A_
2
2
12 D(x, y) (1, 2)
13 Draw a horizontal line from A___
to C. For
ABC, the length of the base AC is the difference between the x-coordinates of A and C;
3 (7) 10. The
___ height is the vertical distance from B to AC or the difference between
the y-coordinates; 5 1 4. The area of
1 (10)(4) 20. For
1 bh _
ABC is A _
2
2
ADC, the base is 10 and the height is 4. The
1 bh _
1 (10)(4) 20.
area of ADC is A _
2
2
The area of quadrilateral ABCD is 20 20 40.
Chapter 6: Transformations and the Coordinate Plane
14 The distance from A to B is
2
2
[2 (7)] (5 1) √
2
5
4 2 41 . The distance from B to C
2
is √
[3 (2)] (1 5)2 2
5
4 2 41 . The distance from C to D
[(2) 3]2 [(3) 1]2 is √
2
5
4 2 41 . The distance from D to A
is √[(7) (2)] 2 [(1 (3)] 2
2
2
5 4 16
17
18
19
20
21
22
41 . The perimeter of ABCD
41 41 41___
41 4 41 .
The length of the base AC is the difference
between the x-coordinates of A and C;
6 (4) 10. The
___ height is the vertical distance from B to AC or the difference between
the y-coordinates; 6 (6) 12. Therefore,
1 (10)(12) 60.
1 bh _
A_
2
2
Using the distance formula, AB BC 13
and CA 10. The perimeter of ABC 13 13 10 36.
Isosceles because PA AT 5
53 , CB √
74 , and
Scalene because AB √
29
AC √
130 , AD √
26 ,
Scalene because MA √
104
and MD √
26 , IT √
125 , and
Scalene because WI √
109
WT √
All three sides measure 2, therefore the triangle must be equilateral.
The radius is the distance from the center to
any point on the circle.
2
2
1) (2 2) 5
r √(4
15
Diameter 2(5) 10
23 The two diagonals are congruent because
AD CB 2.
2
2
2
24 AB [(1) 3] (2 2) 32
2
2
2
BC (3 1) (2 4) 8
2
2
2
AC [(1) 1] [(2) 4] 40.
2
2
2
32 8 40 or AB BC AC .
5
6
7
8
9
10
11
12
13
14
15
16
17
18
T 0, 2
T 3, 0
(4, 1)
(3, 7)
(2, 4)
(4, 3)
(4, 2)
(2, 2)
T 3, 2
T 2, 2
(3, 0)
(7, 3)
(4, 7)
K(8, 5) → (2, 9), E(10, 3) → (4, 1),
N(2, 2) → (8, 6)
19
y
10
9
8
7
D"
6
D
E"
E
5
4
W"
3
2
D'
E'
W
1
5 4 3 2 1
1
1
2
3
4
5
6
7
8
9 10
x
W'
2
3
4
5
a D’(1, 2), E’(6, 3), W’(2, 1)
b D(0, 6), E(7, 7), W(3, 3)
c T 3, 1
20
y
10
9
8
7
6
S'
S"
5
4
A'
3
2
W'
1
5 4 3 2 1
1
2
3
S(3, 4)
H'
A"
A(0, 1)
1
2
3
4
W(1, 2)
H"
H(5, 1)
5
6
7
8
9 10
x
W"
4
5
a W’(2, 0), A’(3, 3), S’(0, 6), H’(2, 3)
b W(5, 1), A(4, 2), S(7, 5), H(9, 2)
c T 4, 1
6-2 Translations
(pages 94–95)
1 (1) (1, 2)
2 (2) (3, 2)
3 (4) T 8, 4
4 (4) 6
6-2 Translations
29
6-3 Line Reflections and
Symmetry
(pages 100–101)
1 (3) A
2 (3) V
3 (1) only vertical line symmetry
4 (2) 2
5 (3) (4, 7)
6 y-axis
7 both
8 x-axis
9 y-axis
10 P’(1, 8)
13
14
15
16
(5, 2)
(2, 3)
(2, 5)
A’(2, 1), B’(3, 4), C’(4, 5)
y
8
7
6
C(4, 5)
5
3
2
5
5
6
7
8
x
B'(3, 4)
5
6
7
8
y
6
I'(–1, 4) 5 I(1, 4)
1
1
2
3
4
5
6
2
7
8
9 10
x
4
H'(–4, 3)
2
5
J'(5, 2)
1
6 5 4 3 21G
1
4
H(4, 3)
3
J(–5, 2)
y 2
3
G'1
2
3
4
5
x
6
2
6
P'(1, 8)
18 A’(3, 3), B’(5, 8), C’(1, 5)
9
y
10
10
11 P’(3, 7)
9
B'(5, 8)
8
y
7
6
8
B(8, 5)
C'(1,5 5)
7
4
6
3 A'(3,
5
2
4
6 5 4 3 2 1
1
1
2
3
4
5
6
7
8
x
x
y
1
8 7 6 5 4 3 2 1
1
2
3)
A(3, 3)
1
3
2
1
2
3
4
C(5, 1)
5
6
7
8
9 10
x
2
3
4
y 2
3
19 Q’(2, 4), Z’(3, 4), D’(1, 2)
4
5
y
6
7
6
8
5
Q'(2, 4)
Z'(3, 4)
4
Z(4, 3)
12 P’(4, 1)
3
2
y
1
8 7 6 5 4 3 2 1
1
8
7
Q(4, 2)
6
5
3
2
P'(4, 1)
1
8 7 6 5 4 3 2 1
1
2
3
4
5
1
2
3
4
5
6
7
8
x
y 2
P(4, 5)
6
7
8
Chapter 6: Transformations and the Coordinate Plane
D'(1, 2)
1
D
(2, 1)
3
4
4
30
4
A'(2, 1)
4
C'(4, 5)
2
P'(3, 7)
3
3
3
P(3, 3)
2
2
P(2, 4)
4
8
1
17 G’(0, 0), H’(4, 3), I’(1, 4), J’(5, 2)
6
7
A(2, 1)
1
8 7 6 5 4 3 2 1
1
y
6 5 4 3 2 1
1
B(3, 4)
4
2
3
y
4
5
x
6
7
8
x
20
6-5 Rotations
y
6
y
4
2
yx
5
x
3
2
1
6 5 4 3 2 1
1
2
3
1
2
3
4
5
6
x
x y2
4
5
6
6-4 Point Reflection and
Symmetry
(pages 103–104)
1 (2) H
2 (1) only line symmetry
3 (3) both point and line symmetry
4 (3) (k, 2k)
5 line symmetry
6 both
7 point symmetry
8 both
9 (1, 3)
10 (3, 1)
11 (9, 0)
12 (2, 6)
13 (4, 4)
14 (8, 2)
15 (2, 4)
16 (2, 8)
17 (10, 10)
18 Point reflection is a transformation of a
figure into another figure.
(page 107)
1 (2) trapezoid
2 (1) regular hexagon
3 (2) regular pentagon
4 (2) rectangle
5 (1) equilateral triangle
6 (1)
7 Z
8 X
9 Y
10 Z
11 (1, 5)
12 (3, 3)
13 (2, 8)
14 (2, 3)
15 (2, 2)
16 (4, 4)
17 (6, 3)
18 (2, 6)
19 (5, 5)
20 W(3, 2), H(8, 2), Y(5, 10)
6-6 Dilations
(page 110)
1 (3) (4, 10)
1
2 (2) _
2
3 (4) (x, y) → (2x, 2y)
4 (4) 20 square inches
5 (6, 9)
9
6 3, _ or (3, 4.5)
2
7 (4, 6)
(
)
For problems 8–11, see figure below.
Point symmetry is a quality of a figure. A figure with point symmetry is not changed by a
reflection through that point. (It is rotated in
either direction 180 about a point.)
8 A(0, 8), B(2, 2), C(6, 4)
3
3
1 , 1_
1
9 A(0, 3), B _, _ , C 2_
2
4
4
4
10 A(0, 12), B(3, 3), C(9, 6)
(
) (
)
6-6 Dilations
31
11 A(0, 8), B(2, 2), C(6, 4)
10
y
8
y
7
10
6
9
5
8
4
7
3
2
6
5
(8)
4
1
A
8 7 6 5 4 3 2 1
1
2
C
B
2
1
2
3
4
5
6
7
8
9 10
x
5
Z'(3, 4)
6
7
x
8
M(5, 2)
7
8
(11)
(10)
9
direct isometry
11
y
8
10
7
11
6
12
5
13
4
14
13
5
6
8
12
4
Z(3, 4)
5
(9)
6
7
3
4
3
4
2
3
1
10 9 8 7 6 5 4 3 2 1
1
1
K'(2, 2)
K(2, 2)
2
M'(5, 2)
3
3
K'(1, 1) 2
(1, √3 )
8 7 6 5 4 3 2 1
1
1
2
3
4
5
Z'(0, 1)
2
K(2,
2)
( √_22 , √_22 )
14 (3a, 3b)
1 ; (2, 4) → (1, 2)
15 k _
3
1 ; (2, 4) → (1, 2)
16 k _
2
17 (3, 12)
18 G(1, 1), N(4, 1), A(4, 1), T(1, 1)
M'(2, 1)
1
7
8
x
M(5, 2)
3
4
6
Z(3, 4)
5
6
7
8
direct isometry
12
y
5
4
3
2
1
5 4 3 2 1
1
6-7 Properties Under
Transformations
(pages 111–113)
1 (2) dilation
2 (4) dilation
3 (2) translation
4 (2) r y x
5 (3) dilation D 1
6 r y-axis
7 T 5, 0
8 R 270 or R 90
9 T 4, 3
1
2
3
4
5
2 2)
K(2,
5
7
8
9 10 11 12
x
M(5, 2)
3
4
6
K'(4, 4) W'(10, 4)
Z(3, 4)
6
7
8
Z'(6, 8)
9
10
not an isometry
13
y
10
9
8
A" R'''
7
6
5
C"
A'''
R"
4
3
C'''
2
1
10 9 8 7 6 5 4 3 2 1
1
C
1
2
3
4
5
2
3
4
5
6
7
8
9 10
R
R'
A
C'
6
7
8
A'
9
10
a C(3, 5), A(4, 7), R(7, 4)
32
Chapter 6: Transformations and the Coordinate Plane
x
b C(3, 5), A(4, 7), R(7, 4)
c C(5, 3), A(7, 4), R(4, 7)
d c
y
14
12
11
D
10
I
9
8
D"
D'
7
6
K
5
D'''
4
3
2
I"
I'
1
K"
K'
11 10 9 8 7 6 5 4 3 2 1
1
I'''
1
2
3
4
5
6
7
8
9 10 11
x
13 A
14 R 90
15 Any combination of three rotations, the sum
of whose angles is 100
16 Any combination of two rotations, the sum
of whose angles is 180
17 Glide reflection
18 (6, 1)
19 (2, 3)
20 (2, 7)
y
21
10
2
9
4
7
K''' 3
4
2
C
1
C'
10 9 8 7 6 5 4 3 2 1
1
1
2
3
4
5
6
7
8
9 10
x
C"2
y
3
4
9
5
M"
8
7
6
7
A"
6
P"
5
4
2
J"
J''' 1 J'
1
2
3
4
5
J
2
3
4
A'
5
8
9
10
R"
3
10 9 8 7 6 5 4 3 2 1
1
P
3
10
A'''
yx
M'
5
a K(6, 1), I(9, 2), D(10, 7)
b K(6, 1), I(9, 2), D(10, 7)
c K(1, 2), I(4, 1), D(5, 4)
R'''
M
6
5
15
P'
8
6
7
8
9 10
C(2, 1), M(7, 5), P(4, 8)
C(2, 1), M(7, 5), P(4, 8)
d r y x
x
A
R'
R
22
y
10
a J(1, 0), A(3, 1), R(2, 2)
b J(0, 2), A(2, 6), R(4, 4)
c J(2, 0), A(6, 2), R(4, 4)
9
8
Y'
7
Y
6
X"
5
X
4
X'
3
2
6-8 Composition of
Transformations
(pages 117–119)
1 (3) R 200
___
2 (2) AT
3 (1) (x, y)
4 (3) r y x D 3
5 (1) a direct isometry
6 (2) (x, y)
7 (4) (3, 8)
8 (4) D
9 (3) T 4, 0
10 (1) (x, y)
11 (3) orientation
12 N
Z'
1
10 9 8 7 6 5 4 3 2 1
1
Z
1
2
3
4
5
2
6
7
8
9 10
x
Y"
3
4
5
yx
6
7
8
Z"
9
10
X(5, 3), Y(2, 6), Z(7, 1)
X(3, 5), Y(6, 2), Z(1, 7)
d (1) rotation
Chapter Review (pages 119–121)
1 (1) I
2 (1) WOW
3 (3) parallel to the y-axis
4 (3) (5, 2)
Chapter Review
33
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
(3) (x, y) → (x, 2y)
(1) translation
(1) rotation
(1) A
(4) D
(2) (5, 4)
(3) reflection in the line y x
(2) (x, y) → (4x, 2y)
(4) y 0
(3) r x 1 r y-axis
1
(1) _
2
(4)
(3) (6, 1)
5
13
10
2 √
13
10
3 √
(1, 1)
(2, 6)
(3, 0)
(4, 1)
r y x(1, 8) (8, 1)
28 (4, 9)
29 a (2, 2)
b (4, 0)
c (2, 2)
d (2, 2)
e (3, 2)
30 a and c
y
H
10
T
8
6
4
C
2
14
12
10
8
6
4
2
T'(9, 1)
2
4
6
2
4
C'(4, 3)
8
10
T"(8, 2)
12
H'(10, 3)
6
8
C"(2, 10)
10
H"(10, 10)
b ry x
CHAPTER
Polygon Sides and
Angles
7-1 Basic Inequality
Postulates
(pages 125–126)
Note: Since there are many variations of proofs,
the following is simply one set of acceptable
statements to complete each proof. Depending
on the textbook used, the wording and format
of reasons may differ, so they have not been
supplied for the method of congruence applied
34
Chapter 7: Polygon Sides and Angles
14
7
in each problem. (These solutions are intended
to be used as a guide—other possible solutions
may vary.)
1 1. m1 m2
2. m2 m1
3. m2 m3
4. m3 m2
5. m3 m1 (Transitive postulate of
inequality)
6. m1 m3
x
2 1. PQ
___ PS
−−
2. PQ PS
3. 1 2
4. m1 m2
5. m1 m3
6. m2 m3
CE BD
CF BF
CE CF FE
CF FE BD
BD BF FD
CF FE BF FD
BF FE BF FD
BF FE BF BF FD BF
or
___FE ___FD
4 1. AE EB
2. ABE BAE
3 1.
2.
3.
4.
5.
6.
7.
8.
3.
4.
5.
6.
7.
8.
9.
10.
5 1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
12.
(Isosceles triangle theorem)
(Substitution
postulate of
inequality)
(Subtraction
postulate of
inequality)
(Isosceles triangle theorem)
mABE mBAE
mDAB mCBA
mDAB mCAD mBAE
mCAD mBAE mCBA
mCBA mABE mDBC
mCAD mBAE
mABE mDBC
mCAD mABE
mABE mDBC
mCAD mABE
(Subtraction
mABE mABE
postulate of
mDBC mABE
inequality)
or mCAD mDBC
−−
C is the midpoint of AB.
AC CB
AB AC CB
AB AC AC
or AB 2AC
AB AC
_
2
−−
F is the midpoint of DE.
DF DF
DE DF FE
DE DF DF
or DE 2DF
DE DF
_
2
AC DF
AB DF
_
2
AB _
DE
13. _
2
2
AB 2 _
DE 2
14. _
2
2
or AB DE
(Multiplication
postulate of
equality.)
6 1. RP 3RS
3RS
RP _
2. _
3
3
RP
_
RS
or
3
3. RQ 3RT
RQ
3RT
4. _ _
3
3
RQ
_
RT
or
3
5. RS RT
RP RT
6. _
3
RQ
RP
_
_
7.
3
3
RQ
RP 3 _
3
8. _
3
3
or RP RQ
7 1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
12.
8 1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
(Multiplication
postulate of
equality.)
−−
I is the midpoint of EH.
EH EI IH
EI IH
EH EI EI
EH 2EI
−−
J is the midpoint of EF.
EF EJ JF
EJ JF
EF EJ EJ
or EF 2EJ
EH EF
2EI EF
2EI 2EJ
EI EJ
(Division postulate
of equality)
___
BD bisects ABC.
ABD DBC
(Definition of
bisector)
mABD mDBC
mABC mABD mDBC
mABC mABD mABD
or mABC 2mABD
−−
BD bisects ADC.
ADB BDC
mADB mBDC
mADC mADB mBDC
mADC mADB mADB
or mADC 2mADB
7-1 Basic Inequality Postulates
35
11. mABC mADC
12. 2mABD 2mADB
13. mABD mADB
(Division postulate of equality)
1 AC
9 1. AE _
3
1 AC 3 or 3AE AC
2. AE 3 _
3
(Multiplication postulate of equality)
1 AB
3. AD _
3
1 AB 3
4. AD 3 _
3
or 3AD AB
5. AC AB
6. 3AE 3AD (Substitution postulate)
7. AE AD (Division postulate of
inequality)
___
___
A
to
BC.
10 1. AE is the median from
___
2. E is the midpoint of BC.
(Definition of median)
3. BE EC (Definition of midpoint)
4. BC BE EC
5. BC BE BE
or BC 2BE
___
___
6. CD is the median from
___C to AB.
7. D is the midpoint of AB.
(Definition of median)
8. AD DB
9. AB AD DB
10. AB AD AD
or AB 2AD
11. AB BC
12. 2AD 2BE
13. AD BE (Division postulate of
inequality)
7-2 The Triangle Inequality
Theorem
(page 128)
1 Yes
2 No
3 Yes
4 No
5 Yes
6 No
36
Chapter 7: Polygon Sides and Angles
Yes
Yes
Yes
Yes
1
S
9
6 S 10
2.5 S 5.5
0 S 12
1
1 S 3_
15 1_
2
2
7
8
9
10
11
12
13
14
7-3 The Exterior Angles
of a Triangle
(pages 130–132)
1 (2) isosceles
2 (4) right
3 a CAD
b ABC and ACB
c mCAD mABC; mCAD mACB
4 a True
b True
c True
d True
5 6 and 9
6 5, 9, 8, and 10
7 7 and 10
Note: Since there are many variations of proofs,
the following is simply one set of acceptable
statements to complete each proof. Depending
on the textbook used, the wording and format
of reasons may differ, so they have not been
supplied for the method of congruence applied
in each problem. (These solutions are intended
to be used as a guide—other possible solutions
may vary.)
8 1. m1 m3
(Exterior angle
theorem)
2. m3 m5
(Exterior angle
theorem)
(Transitive postu3. m1 m5
late of inequality)
___
___
9 1. AB CD
2. 1 is a right angle.
3. AED is a right angle.
4. AED 1
(Right angles are
congruent.)
5. mAED m1
6. mAED m2
(Exterior angle
theorem)
7. m1 m2
10 1. mDCB mCBA
11
12
13
14
(Exterior angle
theorem)
2. mCBA mCBM mMBA
3. mCBA mMBA
4. mDCB mMBA
(Transitive postulate of inequality)
12x 8 (4x 6) (7x 6)
x8
12x 32 (5x 5) (5x 5)
x 11
Let x be mB, then mA is 3x.
3x x 104
x 26
mA 3x 3(26) 78
If the vertex angle of the triangle measures
130, each base angle measures 25 and
each exterior angle at the base measures
180 25 155.
7-4 Inequalities Involving
Sides and Angles of
Triangles
(page 133)
1 ZXY
2 PRQ
___
3 AC
4 GFH
___
5 KL
6 ACB
___
7 YZ
8 mDGE mDEF mGDE, so
mDGE mGDE and DE EG.
9 PQS QSP, but QSP SQR, so
PQS SQR
10 m4 m3 m1, so m4 m1
Chapter Review (pages 133–136)
1 (3) p m p n
2 (3) 28
3 (2) an obtuse angle
4 True. Transitive postulate of inequality
5 True. Additive postulate of inequality
6 True. Postulate of inequality
7 False
Note: Since there are many variations of proofs,
the following is simply one set of acceptable statements to complete each proof. Depending on the
textbook used, the wording and format of reasons
may differ, so they have not been supplied for the
method of congruence applied in each problem.
(These solutions are intended to be used as a
guide—other possible solutions may vary.)
8 1. m2 m3
2. m1 m2
(Exterior angle
theorem)
3. m1 m3
(Transitive postulate of inequality)
9 1. BEST is a parallelogram.
2. BT
___
___ BE
(Definition of a
3. BT ES
parallelogram)
4. BT ES
5. ES BE
(Substitution
postulate of
inequality)
___
___
10 1. GS GD
2. NPS RSP
(Isosceles triangle theorem)
3. mNPS mRSP
4. mISR mIPN
5. mISR mRSP mIPN mRSP
(Addition postulate of inequality)
6. mISP mRSP mIPN mNPS
7. mISP mIPS
(Postulate of
inequality)
11 1. mBCE mDBC
2. mACE mABD
3. mBCE mACE (Addition
postulate
mDBC mABD
inequality)
4. mACB mABC
(Postulate of
inequality)
___
___
12 1. AD DC
2. ACD CAD
(Isosceles triangle theorem)
3. mACD mCAD
4. mDAB mDCB
5. mDAB mACD (Subtraction
mDCB mACD
postulate of
inequality)
Chapter Review
37
6. mDAB mCAD mDCB mACD
7. mCAB mACB
(Postulate of
inequality)
13 1. PQRS is a parallelogram.
2. SP RQ
(Definition of
parallelogram)
3. ST TP RQ
4. ST TP QU RU
or ST RU QU TP
5. TP QU
6. 0 QU TP
(Subtraction
postulate of
inequality)
7. 0 ST RU
(Addition
8. ST RU
postulate of
inequality)
___
14 ST
___
15 AB
___
16 AB
___
17 BC
−−
18 DE
19 AD BD
20 mB 120
Note: Since there are many variations of proofs,
the following is simply one set of acceptable
statements to complete each proof. Depending
on the textbook used, the wording and format
of reasons may differ, so they have not been
supplied for the method of congruence applied
in each problem. (These solutions are intended
to be used as a guide—other possible solutions
may vary.)
___
21 1. AD
___ bisects CAB.
2. BE bisects CBA.
3. mDAB mEBA
4. mDAB 2 mEBA 2
(Multiplication
postulate of
inequality)
5. mCAB 2(mDAB)
(Definition
of bisector)
6. mCBA 2(mEBA)
(Definition
of bisector)
7. mCAB mCBA
38
Chapter 7: Polygon Sides and Angles
22 1. CF AE
2. CF 2 AE 2
3.
4.
5.
6.
7.
8.
23 1.
2.
3.
(Multiplication postlate
___of inequality)
F is the midpoint of CD.
CF 2 CD
(Definition of
midpoint)
CD AE 2
___
E is the midpoint of AB.
AB AE 2
(Definition of
midpoint)
CD AB
(Multiplication postulate of inequality)
AC CB
CB
AC _
_
(Division postulate of
2
2
inequality)
AC
(Definition of
AD _
2
midpoint)
CB
4. AD _
2
CB
5. BE _
2
24
25
26
27
(Definition of
midpoint)
6. AD BE
Let A be the acute angle and let B be
its supplement. mB 180 mA. Since
mA 90, mB 90 and is therefore
obtuse by the subtraction postulate of
inequality.
Let C be the complement of A, and
let D be the complement of B. mC 90 mA and mC mD by the subtraction postulate of inequality.
m1 m3 because an exterior angle is
greater than either nonadjacent interior angle
and m1 m2. So m2 m3 by the
substitution postulate of inequality.
mABC mABD because a whole is
greater than its parts. mBAC mABC
because they are opposite congruent sides
of a triangle. So mBAC mABD by the
substitution postulate of inequality, and
DB DA because the greater side lies
opposite the greater angle.
28 mBDA mACB because a whole is
greater than its parts. mACB mBCA
because they are opposite congruent sides
of a triangle. So mBDA mBCA by the
substitution postulate of inequality, and
AB AD because the greater side lies
opposite the greater angle.
−−
29 Draw diagonal AC, forming two triangles.
In ABC, mBCA mBAC because
they are opposite congruent sides. In
ADC, mACD mCAD because the
greater angle is opposite the greater side.
mBCA mACD mBAC mCAD
or mBCD mBAD by the addition postulate of inequality.
30 Look at the two right triangles formed with
−−
common side EG. mD mF by the subtraction postulate of inequality, so DE EF
because the greater side lies opposite the
greater angle.
31 m2 180 114 66
m1 180 50 66 64
32 x 45 105
x 60
33 y 180 110 70
x 70 130
x 60
34 True
35 False
36 True
37 False
38 False
CHAPTER
8
8-1 The Slope of a Line
(pages 141–144)
1 (2) (0, 4)
2 (2) It has an x-intercept of 2.
3 (4) y 3
4 (3) a y-intercept of 5
5 (3)
6 (2) x 1
5
7 m _; negative
3
8 m 2; positive
3
9 m _; positive
4
Slopes and Equations
of Lines
10 Undefined
11 m 2; negative
3
12 m _; positive
2
13 m 0; zero
1 ; positive
14 m _
2
15 m 3; positive
16 a y 5
b y1
c x 2
d x 10
e y 4.5
8-1 The Slope of a Line
39
ge
17 a _
fd
b Undefined
a
c _
b
d 1
3
18 m _
2
19 y 3
20 y 14
21 y 4
22 x 0
23 x 15
24 x 8
25 x 4
26 a m 1
b m1
27 a–c Not collinear
28 m 4
29 15 feet
1
1
4
_
_
_
−− , m −−− , m −−− 30 a m AB
3 CD
3 AD
11
b Not collinear
2 , m −− is undefined, m −−− 0. The
_
−− 31 m AB
AD
3 BC
−−
triangle is a right triangle because BC is
−−
parallel to the x-axis and CA is parallel to the
y-axis, making these sides perpendicular to
each other.
8-2 The Equation of a Line
(page 147)
1 a No
b Yes
c Yes
2 a y x 6; m 1, b 6
b y 3x 5; m 3, b 5
c y 3x; m 3, b 0
d y 2x 4; m 2, b 4
2
2 x 4; m _
e y _
,b4
3
3
3
3
f y _x 3; m _, b 3
2
2
g y 3x 5; m 3, b 5
9
9
1x_
1, b _
h y _
; m _
2
2
2
2
3 a x-intercept 7, y-intercept 7
b x-intercept 4, y-intercept 8
c x-intercept 3, y-intercept 6
40
Chapter 8: Slopes and Equations of Lines
d
e
f
4 a
b
c
d
5 a
b
c
d
e
f
6 a
b
c
d
e
f
7 a
b
c
d
e
f
x-intercept 2, y-intercept 4
x-intercept 0, y-intercept 0
x-intercept 6, y-intercept 4
y 2x 5
1x 2
y_
2
y 3
y x 4
yx3
y2
1x 5
y_
2
5
_
y x3
2
3
y _x 2
2
3
_
y x5
5
7x 5
y _
2
y 2x 2
y 5x 10
3
y _x 3
2
y 4x 13
6
y _x 6
5
yx1
2x 2
y _
3
y5
y 2x 1
y 2x 11
1
1x _
y _
3
3
8-3 The Slopes of Parallel
and Perpendicular Lines
(page 150)
1 (4) Line l has a negative slope.
2 (1) 5
9
3 a Slope parallel _
7
7
Slope perpendicular _
9
b Slope parallel 0
No slope for perpendicular line
7
c Slope parallel _
8
8
Slope perpendicular _
7
4
5
6
7
2
d Slope parallel _
5
5
Slope perpendicular _
2
9
e Slope parallel _
10
10
Slope perpendicular _
9
f Slope parallel 13
1
Slope perpendicular _
13
a Perpendicular; slopes are negative
reciprocals
b Perpendicular; slopes are negative
reciprocals
c Neither; slopes are neither equal nor
negative reciprocals
d Neither; slopes are neither equal nor
negative reciprocals
e Perpendicular; lines with a slope of 0 are
perpendicular to lines with no slope
f Parallel; slopes are the same
a Yes
b Yes
c No
17
1x _
a y _
4
4
b y 2x 7
1x 4
c y_
5
a Right triangle
b Not a right triangle
10
7
6
(pages 153–154)
1 (2) (1.5, 1)
2 (4) (2.5, 2)
3 (2) (2, 0.1)
4 (1) (8, 0)
5 (4) (17, 20)
6 (4) (13, 13)
7 (5a, 5r)
8 (3x, 5y)
9 (10, 11)
D
5
4
3
G
2
A
1
8 7 6 5 4 3 2 1
1
1
2
3
4
5
6
7
8
x
2
3
4
L
5
6
7
8
11
12
13
A rectangle is formed.
M GL (0, 1), M LA (2, 1), M DA (2, 3),
M GD (0, 3)
(0, 2)
−−
−−
a BD is parallel to EF because their slopes
3
are _.
4
1 BD
b EF _
2
1 (10)
5_
2
55
5
a m_
2
2
b m _
5
5
c M 4, _
2
35
5
d y _x _
6
6
y 4x 9
1x _
4
y_
5
5
(
14
8-4 The Midpoint of a Line
Segment
y
8
15
)
8-5 Coordinate Proof
(pages 163–165)
1 The triangle is an isosceles triangle because
40 .
WI WN √
−−−
−−−
2 Slopes of NA and AQ are negative reciprocals proving these two lines form a right
angle by being perpendicular; therefore,
NAQ is a right triangle.
26 ; the midpoint of the
3 AM CM BM √
hypotenuse is equidistant from all three
vertices.
4 The slopes are negative reciprocals so the
median is also the altitude.
8-5 Coordinate Proof
41
5 BIG is isosceles because it has two con50 ; and BIG is
gruent sides, BI IG √
−−
a right triangle because BI is perpendicular
−−
to IG.
−−−
−−
6 Two sides (QR and PS) have the same slope
−−
−−
2 and the other two sides (OP
of _
and RS)
3
have the same slope of 3. PQRS is a
parallelogram.
7 SAND is a parallelogram because the diagonals bisect each other at the point (1.5, 0.5).
8 LEAP is a parallelogram because all of the
50 and are congruent.
sides measure √
9 ABCD is a parallelogram because the diagonals bisect each other at the point (2.5, 0.5).
5 and 2 √
13
10 a 6 √
b The diagonals meet at their midpoint
(1, 1).
c BETH is a parallelogram because the
diagonals bisect each other.
d BETH is not a rectangle because the
diagonals are not congruent.
11 a NICK is a parallelogram; both pairs of
opposite sides are parallel.
b NICK is not a rhombus; the diagonals are
not perpendicular.
12 The figure KATE is a square. KATE is a rectangle because the diagonals bisect each other
and are the same length. KATE is a square
because two adjacent sides are congruent.
13 ABCD is a rhombus. ABCD is a parallelogram because the diagonals bisect each other.
ABCD is a rhombus because two adjacent
sides are congruent.
−−
−−
14 a Slopes of SU and UE are negative reciprocals proving these two lines form a right
angle by being perpendicular; therefore,
SUE is a right triangle.
b SU 5 and UE 10
15 a NORA is a parallelogram because the
diagonals bisect each other. NORA is a
rhombus because two adjacent sides are
congruent.
b NORA is not a square because it does not
contain a right angle.
16 a JACK is a trapezoid because it has only
one pair of opposite sides parallel;
−−
−−
slope of JA slope of CR 1.
b JACK is not an isosceles trapezoid because
the legs are not congruent.
17 MARY is a trapezoid because it has only
one pair of opposite sides parallel; slope of
−−−
−−
MA slope of RY 0. MARY is an isosceles
trapezoid because the legs are congruent;
AR YM 5.
18 C(a, 0)
19 P(b a, c)
20 E(a, 0), F(a, 2a), G(a, 2a)
21 C(a, b r)
22 M(a c, b)
23 T(a, 2a)
24 R(a b, c)
25 Yes
−−
−−
26 Slope of AC is 1 and the slope of BD is 1.
Since the slopes are negative reciprocals,
−− −−
AC BD.
−−
27 a Midpoint of BC is (a, b).
2
b MA MB MC √a
b2
28 a
y
E(2x, 2y)
T(0, 0)
29
30
31
32
Chapter 8: Slopes and Equations of Lines
x
b A(x, y), B(3x, y), C(2x, 0)
−−
2
2
c The length of median TB is √
9x y .
−−
The length of median EC is 2y. The length
−−−
of median QA is √
9x 2 y 2 .
d TEQ is an isosceles triangle.
c 2 d 2
TA TR _
2
−−
−−
JA and NE are parallel to the x-axis and
each slope is 0; thus they are parallel to each
−−−
−− c
; thus
other. Slope of AN slope of EJ _
b
they are parallel. Therefore, JANE is a
parallelogram.
−−
Midpoint of AC midpoint of
−−
a r, _
s
BD _
2
2
(a b) 2 c 2
TA EM √
√( )
(
42
Q(4x, 0)
)
8-6 Concurrence of the
Altitudes of a Triangle
(the Orthocenter)
(page 168)
1 (4) obtuse
2 (4) at one of the vertices of the triangle
Exercises 3–9: Check students’ graphs.
3 (2.2, 1.2)
5
4 _, 4
3
5 (1, 0)
6 (6, 6.5)
22
7 2, _
3
8 (1, 1)
9 (1, 0.5)
−−
−− 4
3
10 a Slope of AB is _
. Slope of BC is _. Since
3
4
the slopes are negative reciprocals, the
segments are perpendicular and B is the
right angle.
−−
−−
b AB and BC can each be altitudes. B is the
endpoint of the altitude drawn from B
to AC.
(
)
( )
Chapter Review (pages 168–171)
3
1 (2) y _x 4
4
2 (2) (0, 3)
3 (4) (1, 9)
4 (2) y 5x 1
5 (1) y 3x 3
6 No. Sub-in: 3(5) 4(2) 0
7 x-intercept is (2, 0) and y-intercept is (0, 3).
13 .
Distance is √
8 x-intercept is (15, 0) and y-intercept is (0, 3).
Distance is 15.3.
9 a 10
10 B(10, 10)
___
1 . Slope of ___
AC is 2. Their
11 Slope of OB is _
2
1 (2) 1.
product is _
2
12 a (0, 3)
b (4, 0)
c (8, 3)
13 a m 3
b x-intercept is (4, 0)
c y-intercept is (0, 12)
d (3, 3) is not on the line.
()
14 a neither
b vertical
c vertical
d neither
e horizontal f vertical
3
1
b 2
c _
15 a _
2
4
a
2
2
e _
f _
d _
3
3
d
16 a no slope b 0
c 1
d
b
k
7
_
_
d
e ca
f _
2
h
17 a 7
b 2
c 14
d 2
e 6
f 5
g 4
18 a not collinear
b collinear
c collinear
d collinear
e collinear
f collinear
19 a y 3
9
3
b y _x _
2
2
5
1
_
c y x_
3
3
2
_
d y x
3
19
3
e y _x _
2
2
√
20 a AB AC 2___
10
b Midpoint of BC is (2,
___ 1). Slope of the
median from
___ A to BC slope of altitude
from A to BC 1.
21 a (3, 2)
b (2, 3) c (6.5, 8.5)
____
___
1
22 Slope of DR slope of AW _. Slope of
3
___
____
RA
___ is not
____equal to the slope of DW.
RA DW 5
2 and OP 2 √
5
23 a NO 5 √
___
___
1
_
b Slope of EO is . Slope of PN is 3.
3
Slopes of perpendicular lines are negative
reciprocals of each other.
24 A(0, 3), B(4, 2), C(5, 4), and D(1, 3). ABCD
is a parallelogram because both pairs
___ of opposite sides are parallel. Slope of AB slope
___
___
1 . Slope of ___
of CD _
BC slope of DA 6.
4
25 QRST is a parallelogram because both pairs
of opposite sides are parallel. Slope of
___
___
b . Slope of ___
QR slope of ST _
RS slope
a
___
of TQ 0.
26 a–b Check students’ graphs.
c (2, 10)
Chapter Review
43
27 a M ___
(5, 1), M ___
(4, 3), M ___
TK (2, 1)
KA
AT
___
___
2
b m KA 1, m AT 0, m ___
TK
___
c Slope of line perpendicular to KA is 1.
There ___
is no slope for the line perpendicular to AT. Slope of the line perpendicular
___
1.
to TK is _
2
___
d Perpendicular bisector of KA
___ : y x 6
Perpendicular bisector of AT: x 4
___
1x
Perpendicular bisector of TK: y _
2
e (4, 2)
28 a y 2x 5
b (6, 8)
1x 5
c y_
2
CHAPTER
Parallel Lines
9-2 Proving Lines Parallel
(pages 177–179)
1 none
2 a b, c d
3 none
4 bc
5 ab
6 ac
7 bc
8 lm
9 a and b
10 c and d
11 a and b
Note: Since there are many variations of proofs,
the following is simply one set of acceptable
statements to complete each proof. Depending
on the textbook used, the wording and format
of reasons may differ, so they have not been
supplied for the method of congruence applied
in each problem. (These solutions are intended
to be used as a guide—other possible solutions
may vary.)
44
Chapter 9: Parallel Lines
9
12 1. 1 3
2. 2 4
3. 1 2 3 4
(Addition postulate of equality)
4. ABG DEG
−− −−
5. ED BA (If two lines are cut by a transversal forming a pair of congruent alternate interior angles,
___ the two lines are parallel.)
___
CD
13 1. AF ___
−−
2. FC FC ___ ___
−− −−
3. AF
___ FC
___ CD FC
DF
4. AC ___
−−
5. BC
___ EF
___
6. BC AD
7. BCF is a right angle. (Definition of
right angle)
___
___
8. EF AD
9. EFD is a right angle.
10. BCF EFD
(Right angles are
congruent)
11. ABC DEF
(SAS SAS)
12. ___
BAC
EDF
(CPCTC)
___
13. ED AD (Alternate interior angles are
congruent)
___
14 1. BE bisects ABC.
2. 1 EBC
3. ___
m1 mEBC
4. CE bisects DCB.
5. 2 ECB
6. m2 mECB
7. m1 m2 90
8. mEBC mECB 90
(Substitution
postulate)
9. m1 m2 mEBC mECB
10. m1 mEBC 90
11. m2 mECB 90
12. ABC___
and DCB are right angles.
BC
13. BA
___
BC
14. CD
(Two lines perpendicular to the
CD
15. BA
same line are parallel.)
15 1. 1 3
2. m1 m3
3. 2 4
4. m2 m4
5. m1 m2 m3 m4 360
6. m1 m1 m2 m2 360
or 2(m1) 2(m2) 360
7. m1 m2 180
(Division
postulate)
−−− −−
8. AD BC (Interior angles on the same
side of the transversal are
supplementary.)
9. m1 m1 + m4 m4 360
or 2(m1) 2(m4) 360
10. ___
m1___
m4 180
11. AB CD
___
___
CF
and
of
BE.
16 1. D
is
the
midpoint
of
___
___
DF
2. CD
___
___
DE
3. BD
___
___
4. FC FC
5. CBD FDE
(Vertical
angles are
congruent.)
6. CBD FDE
(SAS SAS)
7. ___
EFD
(CPCTC)
___ DCB
8. AC FE (Alternate interior angles are
congruent.)
17 1. 1 2
2. AFC DCF (Supplementary angles
of congruent angles are
congruent.)
___
___
3. EF
___ CB
___
FC ___ ___
4. FC
___ ___
5. EF
___ FC
___ = BC FC
BF
6. EC
___
___
7. AF CD
8. AFB DCE
(SAS SAS)
9. ABF
DEC
(CPCTC)
___ ___
(Alternate interior an10. AB ED
gles are congruent.)
___
___
FC
18 1. AE
___
___
EF
2. EF
___
___
___
___
EF FC EF
3. AE
___
___
4. AF
___
___ CE
5. DE AC
6. ___
DEA___
is a right angle.
7. BF AC
8. BFC is a right angle.
9. DEA BFC
(Right angles are
congruent.)
___
___
10. DE BF
11. ______
AFB CED
(SAS SAS)
12. AEFC
13. BAF
DCE
(CPCTC)
___ ___
14. AB DC
9-3 Properties of Parallel
Lines
(pages 182–185)
1 (1) same-side exterior angles
2 (2) 2, 3, 6, 7, 10, 11, 14, 15
3 m1 m4 m6 m7 60;
m2 m3 m5 120
4 m1 m4 m5 m8 135;
m2 m3 m6 m7 45
5 ma md mg 65;
mb mc me mf 125
6 w y 70; x z 110
7 a 8x 6x 30 180
x 15
b 2x 10 5x 47
x 19
m3 2(19) 10 48
9-3 Properties of Parallel Lines
45
106
mA 75, mC 67
57
60
1. Perpendicular lines form right angles.
2. Right angles measure 90.
3. When two parallel lines are cut by a
transversal, corresponding angles are
congruent.
4. Congruent angles have equal measure.
5. Transitive property of congruence (3, 5)
6. If an angle measures 90, it is a right
angle.
7. If two lines intersect to form a right angle,
they are perpendicular.
Note: Since there are many variations of proofs,
the following is simply one set of acceptable
statements to complete each proof. Depending
on the textbook used, the wording and format
of reasons may differ, so they have not been
supplied for the method of congruence applied
in each problem. (These solutions are intended
to be used as a guide—other possible solutions
may vary.)
___
13 1. AB
2. 2 is supplement of 1.
3. 2 is supplement of 3.
4. 1 3 (If two angles are supplements
of the same angle, then they
are congruent.)
___ ___
5. BC AD (When two lines are cut by
a transversal creating corresponding angles, the lines are
congruent.)
14 1. ABC
___
___
2. AB BC
3. BAC BCA
(Isosceles triangle
theorem)
_____ ___
4. FHD BC
5. FDA BCA
(Corresponding
exterior angles are
congruent.)
_____ ___
6. EHG AB
7. BAC GEC
(Corresponding
exterior angles are
congruent.)
8. ___
GEC____
FDA
9. HE HD
10. EHD is isosceles. (Definition of
isosceles triangle)
8
9
10
11
12
46
Chapter 9: Parallel Lines
___
___
15 1. DE AC
2. 1 2
3. 3 4
4. 2 3
5. 1 4
___
(Corresponding exterior
angles are congruent.)
(Transitive property of
congruence)
___
BD at E.
16 1. AC
___ intersects
___
2. AE ED
3. A
D
___ ___
4. AD BC
5. A C
(Alternate interior angles are
congruent.)
6. B D
(Alternate interior angles are
congruent.)
7. B C
(Transitive property of
congruence)
___
___
(Isosceles triangle
8. AE CE
theorem)
___
___
9. BE
___ DE
___
(Addition postulate)
10. AC BD
17 1. n m
2. ABE and BAD are supplementary.
(Two interior angles on the
same side of the transversal
are supplementary.)
3. mABE mBAD 180
1 mBAD 90
1 mABE _
4. _
2
2
(Division postulate)
___
5. BC bisects ABE.
1 mABE
6. mABC _
2
(Definition of angle bisector)
___
7. AC bisects BAD.
1 mBAD
8. mACB _
2
(Definition of angle bisector)
9. mABC mACB 90
10. mBCA mABC mACB 180
11. mBCA 90
12. BCA
is a right angle.
___
___
(Perpendicular lines form
13. BC AC
right angles.)
18 1. r m and a b
2. 4 and 3 are supplementary.
3. 2 and 3 are supplementary.
4. (a) 4 2 (Supplements of the same
angle are congruent.)
5. 4 and 5 are supplementary.
19
20
21
22
6. 2 and 1 are supplementary.
7. (b) 5 1 (Supplements of the same
angle are congruent.)
a (x 12) (3x) 180
x 42
m1 m3 m5 54
m2 m4 m6 126
b (2x) (2x 20) 180
x 50
m1 m3 m5 100
m2 m4 m6 80
c (7x 65) (5x 5)
x 30
m2 m5 145
m1 m3 m4 m6 35
a 5
b 7
c 6 and 8
d m5 60
e m4 90
Since the corresponding angles are congruent, the halves of each are congruent. They
form new congruent corresponding angles
cut by the same transversal. The bisectors are
therefore parallel.
Draw a diagonal line, forming two congruent triangles by SAS. The other two sides of
the quadrilateral and remaining angles are
congruent by CPCTC. Therefore, since the
diagonal is a transversal for these sides, the
sides are parallel.
9-4 Parallel Lines in the
Coordinate Plane
(pages 187–188)
1 a 4
7
b _
3
2
c _
5
x
_
d a
2 a 1
3
b _
5
2
c _
5
1
3 a _
4
b 7
c 1
3
d _
2
4 a perpendicular
b parallel
c perpendicular
d parallel
e perpendicular
f neither
5 a 1
5
b _
2
3
c _
10
7
d _
9
4
_
e
3
3
_
f
2
6 a 7
b 1
c 11
d no slope
e 0
1
f _
2
___
1 , ___
7, slope ___
_
AB CD
7 a Slope ___
AB
CD
7
___
___
_
___
___
b Slope AB 2, slope CD 1 , AB CD
2
8 y 5x 3
1x 3
9 y=_
5
3
7
10 y _x _
2
2
1
_
11 y x 3
3
12 y 4
13 y 2x 6
2x 9
14 y _
3
15 k = 1
16 (7, 5)
3
4 , slope ___ _
17 Slope ___
_
. The slopes are
AB
CD
3
4
−−−
−−
negative reciprocals, therefore AB and CD
are perpendicular and ABC is a right
triangle.
18 Opposite sides have the same slope.
1 and m ___ m ___ 5
m ___
_
m ___
PQ
RS
PS
QR
3
19 H(5, 6)
20 a D(4, 3)
9-4 Parallel Lines in the Coordinate Plane
47
9-5 The Sum of the
Measures of the Angles
of a Triangle
(pages 193–195)
1 (4) scalene
2 (3) obtuse
3 (3) 120
4 (2) right
5 (3) 112
6 a base angles are 30, vertex angle is 120
b base angles are 60, vertex angle is 60
c base angles are 52, vertex angle is 76
d base angles are 36, vertex angle is 108
e base angles are 20, vertex angle is 140
7 a base angles are 50, exterior angle is 130
b base angles are 40, exterior angle is 140
c base angles are 54, exterior angle is 126
d base angles are 60, exterior angle is 120
e base angles are 80, exterior angle is 100
8 base angles are 74, vertex angle is 32
9 mP 27, mQ 45, mR 108
10 base angles are 28, vertex angle is 124.
11 exterior angle is 30, base angles are 15,
vertex angle is 150
12 18, 54, 108
13 99, 45, 36
14 mc 35
15 mB 78
16 mx 150
17 mx 30
18 m1 150
19 mD 20
20 md 125
Note: Since there are many variations of proofs,
the following is simply one set of acceptable
statements to complete each proof. Depending
on the textbook used, the wording and format
of reasons may differ, so they have not been
supplied for the method of congruence applied
in each problem. (These solutions are intended
to be used as a guide—other possible solutions
may vary.)
21 1. ___
A C
2. BD bisects ABC.
3. ABD
CBD
___
___
4. BD BD
5. ABD CBD
(AAS AAS)
48
Chapter 9: Parallel Lines
ADB CDB
(CPCTC)
ADB is supplementary to CDB.
ADB and CDB are right angles.
−− −−
BD AC
(Segments forming right
angles are perpendicular.)
22 1. A
C
___
___
2. BD AC
3. BDA and BDC are right angles.
4. BDA
BDC
___
___
5. BD BD
6. BDA DBC
(AAS AAS)
7. ABD
CBD
(CPCTC)
___
(Definition of angle
8. BD bisects ABC.
bisector)
23 Both ABC and DEC share C. Since the
remaining angles are the same as well, 1
and B are___
congruent
corresponding angles.
___
Therefore, BA ED.
24 Compare ABC and DEC. Both are right
triangles with one pair of congruent acute
angles. Therefore, the other pair of acute
angles are also congruent, B CED. But
CED 1, vertical pairs. Then B 1
by the transitive postulate of congruence.
6.
7.
8.
9.
9-6 Proving Triangles
Congruent by Angle,
Angle, Side
(pages 198–200)
1 b and f
2 a Not sufficient. If the third side is congruent, SSS. If included angles are congruent,
SAS.
b Sufficient, SSS
c Sufficient, hypotenuse-angle
d Not sufficient. If either pair of corresponding angles are congruent, AAS.
e Sufficient, AAS
f Not sufficient. If any pair of corresponding sides are congruent, AAS.
g Sufficient, hypotenuse-leg
h Sufficient, SAS
i Sufficient, SSS
j Sufficient, AAS
Note: Since there are many variations of proofs,
the following is simply one set of acceptable
statements to complete each proof. Depending
on the textbook used, the wording and format
of reasons may differ, so they have not been
supplied for the method of congruence applied
in each problem. (These solutions are intended
to be used as a guide—other possible solutions
may vary.)
___ ___
3 1. AB EF
2. ABC
FED
___
___
3. BC DE
4. 1 2
5. I II
(ASA ASA)
4 1. 1 4
2. ___
B ___
D
3. AC AC
4. I
II
(AAS AAS)
___ ___
5 1. AB DE
2. DEC BAC (Alternate interior
angles are congruent.)
3. ABC EDC (Alternate interior
angles
___ are congruent.)
BD.
4. C
is
the
midpoint
of
___
___
(Definition of
5. BC CD
midpoint)
6. ABC EDC (AAS AAS)
6 1. B D
2. BEC
DEA
___
___
3. BC AD
4. AED
CEB (AAS AAS)
___
___
(CPCTC)
5. AE CE
7 1. A
E
___ ___
DC
2. BC
___
___
CE
3. AC
___
___
BC (Subtraction
4. AC
___
___
CE___
DC___
postulate)
AB
DE
or
___
___
5. BG AE
6. BGA is a right (Definition of
angle.
right angle)
___
___
AE
7. DF
____
8. DFE is a right angle.
9. BGA DFE (Right angles are
congruent.)
10. BGA
DFE
(AAS
AAS)
___
___
(CPCTC)
11. BG DF
8 1.
2.
3.
4.
5.
9 1.
2.
3.
4.
5.
6.
7.
8.
9.
10 1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11 1.
2.
3.
4.
5.
6.
12 1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
12.
___
___
AB EF
ABC FED
A
F
___ ___
AC DF
I II
(AAS AAS)
___
___
AB
CD
___
___
CD AB
CDA
is a right angle.
___
___
AE CB
AEC is a right angle.
BAC BCA
(Isosceles triangle
theorem)
___
___
AC AC
CDA___
AEC
(AAS AAS)
___
CD AE
(CPCTC)
___
AC.
E
is
the
midpoint
of
___
___
AE
CE
___ ___
AF BD
AFE
is a right angle.
___
___
CD BD
CDE is a right angle.
AFE CDE
CED AEF
(Vertical angles are
congruent.)
AFE
CDE
(AAS AAS)
___
___
AF
CD
(CPCTC)
___
___
QR SR
RQS RSQ
(Isosceles triangle
theorem)
1
2
___
___
QS QS
QTS
(AAS AAS)
____
___SMQ
QM
ST
(CPCTC)
___
___ BA
CD
___
___
BA AD
BAQ
is a right angle.
___
___
CD AD
(Two lines perpendicular to the same line
are parallel.)
CDP is a right angle.
BAQ CDP
B
C
___ ____
AP
___ QD
___
PQ
___ PQ
___
AD PD
(Addition postulate)
BAQ
CDP
(AAS
AAS)
___
___
BA CD
(CPCTC)
9-6 Proving Triangles Congruent by Angle, Angle, Side
49
9-7 The Converse of the
Isosceles Triangle Theorem
(pages 202–203)
1 2(3x 4) 2x 4 180
x 22
mA mC 70
mB 40
2 2(3x 3) 4x 16 180
x 19
mA mB mC 60
3 2x 180 82 82
x8
4 (2x 14) (3x 2) (5x 8) 180
x 16
2x 14 2(16) 14 46
3x 2 3(16) 2 46
5x 8 5(16) 8 88
Since two angles are equal in measure,
ADC is isosceles.
5 mA mB 49
mx 180 49 49
my 49
mz 180 82 98
6 mQ 58
mx mz 64
my 116
Note: Since there are many variations of proofs,
the following is simply one set of acceptable
statements to complete each proof. Depending
on the textbook used, the wording and format
of reasons may differ, so they have not been
supplied for the method of congruence applied
in each problem. (These solutions are intended
to be used as a guide—other possible solutions
may vary.)
7 1. A
C
___ ___
AB
2. BC
___
___
3. AD EC
4. ABD
CBE
(SAS SAS)
___ ___
8 1. BD AC
2. 1 ACB
3. 2 BAC
4. 1 2
5. ACB BAC
(Substitution
postulate)
6. ABC is isosceles. (Definition of
isosceles triangle)
50
Chapter 9: Parallel Lines
9 1. 1 2
2. 3 4
___
___
3. AB BC
4. ABC is isosceles.
___
___
10 1. BD
___ AE
___
2. AC CE
3. A E
4. A B
(Supplements of
congruent angles are
congruent.)
(Definition of
isosceles triangle)
(Two parallel lines are cut
by a transversal then the
corresponding angles are
congruent.)
5. E D (Two parallel lines are cut
by a transversal then the
corresponding angles are
congruent.)
6. B D (Transitive postulate of
congruence)
___
___
7. BC DC (Congruent angles imply
congruent sides.)
___
___
11 1. BC BD
2. BCD BDC
3. BDA BDE
(Linear pairs of
congruent angles)
4. BAC
BED
(ASA ASA)
___
___
BE
(CPCTC)
5. AB
___
___
12 1. AB EB
2. BAE
BEA
___ ___
3. BD AE
4. BAE 1
5. BAE 2
6. 1 2
(Substitution
postulate)
___
___
SR
13 1. PQ
___ ___
2. PQ SR
3. QPR SRP
(Alternate interior
angles are
congruent.)
___
4. PR bisects QPS.
5. QRP RPS
(Definition of
bisector)
6. SRP RPS
(Transitive postulate
of congruence)
___
___
(Converse of the
7. PS SR
isosceles triangle
theorem)
14 1.
2.
3.
4.
5.
6.
7.
15 1.
2.
3.
4.
5.
6.
7.
8.
___
___
DF FE
2 3
1 4
ADF CEF
(Linear pairs of congruent angles)
ADF CEF
(ASA ASA)
A C
(CPCTC)
ABC is an isosceles triangle. (Definition
of
an isosceles triangle)
______
AEDC
1
2
___ ___
BE BD
AEB
CDB
___
___
AE DC
ABE CDB (ASA ASA)
A C
(CPCTC)
ABC is an isosceles triangle. (Definition
of an isosceles triangle)
9-8 Proving Right Triangles
Congruent by HypotenuseLeg; Concurrence of Angle
Bisectors of a Triangle
(pages 207–208)
1 (a)
2 (d)
3 (c)
4 (b)
5 (a)
6 m1 24, m2 52, m3 104,
m4 52, m5 14, m6 114,
m7 14, m8 24, m9 142
Note: Since there are many variations of proofs,
the following is simply one set of acceptable
statements to complete each proof. Depending
on the textbook used, the wording and format
of reasons may differ, so they have not been
supplied for the method of congruence applied
in each problem. (These solutions are intended
to be used as a guide—other possible solutions
may vary.)
___
___
PR
7 1. SQ
___
___
2. SA PB
3. SQP and SAP are right angles.
4. SQP
and SAP are right triangles.
___
___
5. SP SP
6.
7.
8.
9.
___
___
SQ SA
SQP SAP
SPQ
APS
___
PT bisects RPB.
___
___
(HL HL)
(CPCTC)
(Definition of
angle bisector)
8 1. AE
___ BC
___
2. CD AB
3. CDA and CEA are right angles.
4. CDA
and CEA are right triangles.
___
___
AC
5. AC
___
___
6. AE CD
7. ACE
CAD
(HL HL)
___
___
9 1. AE
___ BC
___
2. CD AB
3. AEB and CDB are right angles.
4. AEB
and CDB are right triangles.
___
___
5. DB EB
6. DBE DBE
7. ___
AEB___
CDB
(Leg–acute angle)
CD
(CPCTC)
8. AE
___
___
AC
10 1. BD
___
___
2. QS PR
3. BDA and QSP are right angles.
4. BDA and QSP are right triangles.
5. ABC
PQR
___
___
(CPCTC)
6. AB PQ
7. A P
(CPCTC)
8. BDA QSP
(Hypotenuse–
acute angle)
9. ABD PQS
(CPCTC)
10. mABD
mPQS
___
11. BD bisects ABC.
1 mABC mABD mPQS
12. _
2
1 mPQR (Division
_
13. 1 mABC _
2
2
postulate)
1
14. mPQS _mPQR (Transitive
2
postulate)
___
(Definition of
15. QS bisects PQR.
angle bisector)
___ ___
___
___
11 1. AB CF, DE CF
2. ABF and CED are right angles.
3. ABF
and CED are right triangles.
___
___
AF
4. CD
___
___
BE
5. BE
___
___
6. CE FB
7. ABF
CED
(HL HL)
___
___
8. AB DE
9-8 Proving Right Triangles Congruent by Hypotenuse-Leg; Concurrence of Angle Bisectors of a Triangle
51
___
___ ___
___
CE BA, BD AC
CEA and BDA are right angles.
CEA
and BDA are right triangles.
___
___
AB AC
ABC ABC
CEA BDA (Hypotenuse–acute
angle)
___
___
(CPCTC)
7. CE BD
12 1.
2.
3.
4.
5.
6.
13 1.
2.
3.
4.
5.
6.
7.
___
___ ___
___
AC BF, ED BF
ACB and FDE are right angles.
ACB
and FDE are right triangles.
___
___
AC
ED
___
___
BA EF
ACB FDE (HL HL)
A E
(CPCTC)
9-9 Interior and Exterior Angles of Polygons
(pages 210–212)
1 Polygon
52
Number
of Sides
Number
of Triangles
Sum of
Interior
Angles
180(n 2)
Measure of
Each Exterior
180(n 2)
Angle
n
Measure of
Each Interior
Angle
_
Triangle
3
1
180(1) 180
180
_
60
3
360
_
120
3
Quadrilateral
4
2
180(2) 360
360
_
90
4
360
_
90
4
Pentagon
5
3
180(3) 540
540
_
108
5
360
_
72
5
Hexagon
6
4
180(4) 720
720
_
120
6
360
_
60
6
Heptagon
7
5
180(5) 900
900
_
128.57
7
360
_
51.43
7
Octagon
8
6
180(6) 1,080
1,080
_
135
8
360
_
45
8
Nonagon
9
7
180(7) 1,260
1,260
_
140
9
360
_
40
9
Decagon
10
8
180(8) 1,440
1,440
_
144
10
360
_
36
10
24-gon
24
22
180(22) 3,960
3,960
_
165
24
360
_
15
24
n-gon
n
n2
180(n 2)
180(n 2)
_
n
360
_
n
Chapter 9: Parallel Lines
2 18,000 180(n 2)
n 2 100
n 102
3 Diagonals n 3
a 0
b 1
c 2
d 4
e 8
4 Triangles n 2
a 7
b 10
c 15
d 98
5 180n
a 720
b 900
c 1,160 d 3,420
sum
_
6
180
a 1,000 sides
b 100 sides
360
_
7 n
a 40
b 36
c 10
d 5
360
8 __
# of degrees
a 12
b 10
c 6
d 8
180(n
2)
9 # of degrees _
n
a 18
b 360
c 8
d 6
10 sum 180(n 2)
a 12
b 24
c 50
d 1,000
360
11 Each exterior angle is 30. _ 12 sides
30
12 180(n 2) 720
180n 360 720
180n 1,080
n6
180(n 2)
8 360
_
13 _
n
n
180(n 2) 2,880
n 2 16
n 18
14 m1 70, m2 75, m3 105,
m4 145
15 m1 90, m2 110, m3 70,
m4 35, m5 95, m6 85
16 m1 m2 m6 120
m3 m4 m5 60
17 180(n 2) 5(360)
n 12
18 (5x 10) (6x 25) (6x 25) (5x 10) (3x 5) 180(3)
25x 65 540
x 19
Interior angles: 105, 139, 139, 105, 52
19 x + 3x + 4x + 4x + 6x 360
18x 360
x 20
Exterior angles: 20, 60, 80, 80, 120
Interior angles: 160, 120, 100, 100, 60
20 (3x 4) (7x 7) (6x 5) (5x 8) (3x 2) 5x 360
29x 348
x 12
Exterior angles: 40, 91, 67, 68, 34, 60
Interior angles: 140, 89, 113, 112, 146, 120
Chapter Review (pages 212–216)
1 a no slope
1
b _
4
15
c _
7
5
d _
2
3
e _
5
f 0
2 a 2
b 1
c no slope
d 0
7
e _
5
f no slope
3
5
3 m ___
_, m ___
_. Slopes are negative
AC
5 BC
3
reciprocals.
m ___
m ___
m ___
1. Opposite sides
4 m ___
LA
SL
BC
PS
are parallel (slopes are equal) and slopes of
consecutive sides are negative
reciprocals (sides are perpendicular).
___
___
___
5 m ___
PL m AN 1 and m PS m LA 4.
Opposite sides are parallel (slopes are
equal).
6 For parallel lines m and n cut by transversal a, m6 m10. So 2y 3y 10, and
y 38.
m1 m6 m9 m14 76
m2 m5 m10 m13 104
For parallel lines m and n cut by transversal b, m4 m12. So 2y 15 3y 7,
and y 22.
m4 m7 m12 m15 59
m3 m8 m11 m16 121
7 m1 42, m2 48, m3 42, m4 42
8 mx 85
9 mx 60
10 mx 68
11 130
Chapter Review
53
mx 45, my 45
mx 98, my 82
mx 60, my 70
mx 65, my 52
mx 67, my 78
mx 15, my 55
mx 55, my 62.5
5
b 20
c 35
n
3
_
d 594
e
2
14 102 sides
15 8 sides
16 14 sides
17 6 sides
18 a mx 85
b my 55
19 mD 45
20 m1 m2 25
21 m1 140
Note: Since there are many variations of proofs,
the following is simply one set of acceptable
statements to complete each proof. Depending
on the textbook used, the wording and format
of reasons may differ, so they have not been
supplied for the method of congruence applied
in each problem. (These solutions are intended
to be used as a guide—other possible solutions
may vary.)
22 1. 6 4
2. x y
3. x z
4. y z
(Transitive postulate)
23 1. 2 4
2. 1 3
3. 1 2 3 4
(Corresponding angles are
4. m r
congruent.)
___ ___
24 1. QR PS
2. QPS QRS
3. RPS QRP
4. QPS RPS QRS QRP
or QRP
___
___ SRP
(Corresponding angles are
5. QP RS
congruent.)
___
25 1. BC bisects ABD.
2. 2 3
3. 1 2
4. 1 3 (Transitive postulate)
5. 5 1
12 a
b
c
d
e
f
g
13 a
54
Chapter 9: Parallel Lines
26
27
28
29
30
31
6. 3 5
7. a b (Alternate interior angles are
congruent.)
___
1. AC bisects BCD.
2. 3
4
___ ___
3. AB BC
4. ABC
is a right angle.
___
___
5. CD AD
6. CDA
is a right angle.
___
___
7. AC AC
8. I II
(Hypotenuse–acute
angle)
1. ABCD
___
___
2. CF DE
3. ACF
BDE
___
___
DE
4. CF
___
___
CD
5. AB
___
___
___
___
BC
6. AB
___
___ CD BC
7. AC BD
8. ACF
BDE
(SAS SAS)
___
___
9. AF
___ BE
1. BE bisects ABC.
2. DBE
EBA
___ ___
3. DE BA
4. DEB EBA
(Alternate interior
angles are congruent.)
5. DEB
DBE
(Transitive postulate)
___
___
6. DB DE
7. BDE is isosceles. (Definition of
isosceles triangle)
___
1. AF
___
___
2. AB
___ CD
___
EF
3. AC
___
___
___
___
CE EF CE
4. AC
___
___
CF
5. AE
___ ___
6. AB CD
7. BAE DCF
(Corresponding angles are congruent.)
8. BAE
DCF
(SAS SAS)
___
___
DF
(CPCTC)
9. BE
___
___
1. AB CB
2. A C
3. B B
4. ABE
CBD
(ASA ASA)
___
___
5. AE
___ CD
___
1. BA AE
2. BAE is a right angle.
3. 1 2
4. B D
5.
6.
7.
8.
9.
32 1.
2.
3.
4.
5.
33 1.
2.
3.
4.
5.
6.
7.
8.
34 1.
2.
3.
4.
5.
6.
7.
35 1.
2.
___
___
AE AE
ABE EDA
(AAS AAS)
BAE DEA
(CPCTC)
DAE
is
a
right
angle.
___
___
DE AE
(Perpendicular lines
intersect to form right
angles.)
___
___
BA DE
CAE
CEA
___
___
AE AE
ABE EDA
(SAS SAS)
BEA DAE
(CPCTC)
1 2
3 4
5 6
(Linear pairs of
congruent angles
are congruent.)
___
___
AD
EC
___
___
DE
DE
___
___
AE DC
(Addition postulate)
ABE CBD
(ASA ASA)
ABE
CBD
(CPCTC)
___
___
___
RS ST TR
1
___ 2
___
RP TP
(Converse of isosceles
triangle theorem)
___
___
SP SP
RPS TPS
(SSS SSS)
RSP
TSP
(CPCTC)
___
SP bisects RST. (Definition of an
___ ___
___ angle bisector)
___
PT QR, RS PQ
RSP and PTR
(Definition of right
are right angles.
angles)
RSP
and PTR are right triangles.
___
___
PT
RS
___
___
PR PR
RSP PTR
(HL HL)
SRP TRP
(CPCTC)
PQR is isosceles.
(Converse of
isosceles triangle
theorem)
___
___ ___
___
1. AS PR, BT PR
2. ASP and BTR are right angles.
3. PAS and RBT are right triangles.
4. 1 2
5. PAS RBT
(Linear pairs of
congruent angles
are congruent.)
___
___
6. AS BT
7. PAS RBT
(Leg–acute angle)
8. APS BRT
(CPCTC)
9. PQR is isosceles.
(Definition of an
isosceles triangle)
True
True
False
True
True
False
True
False
True
False
False
True
True
True
3.
4.
5.
6.
7.
8.
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
Chapter Review
55
CHAPTER
Quadrilaterals
10-2 The Parallelogram
(pages 221–223)
1 mP 48, mQ mS 132
2 mR 30, mQ mS 150
3
mQ mR 180
(3x 5) (4x 25) 180
7x 210
x 30
mS mQ 85
mP mR 95
4
mP mQ 180
(7x 12) (2x 3) 180
9x 189
x 21
mR 135
mS 45
1 (5x 3)
5 3x 5 _
2
6x 10 5x 3
x7
PR 5(7) 3 32
QS 3(7) 5 16
6 PRS PRQ; QPS SRP; QMP SMR; QMR SMP
7 x (5x 6) 180
x 31
The angles are 31 and 149.
8 (3x 24) x 180
x 39
The angles are 39 and 141.
9 x3
10 4
11 mETS mBTE mTEB 42
56
Chapter 10: Quadrilaterals
10
Note: Since there are many variations of proofs,
the following is simply one set of acceptable
statements to complete each proof. Depending
on the textbook used, the wording and format
of reasons may differ, so they have not been
supplied for the method of congruence applied
in each problem. (These solutions are intended
to be used as a guide—other possible solutions
may vary.)
12 1. ABCD ___
and DAPQ are parallelograms.
−−−
(Opposite sides of
2. AD BC
a parallelogram are
congruent.)
−−− −−
(Opposite sides of
3. AD PQ
a parallelogram are
congruent.)
−− −−
(Transitive property)
4. BC PQ
13 1. ABCD is a parallelogram.
−− −−−
2. BC AD
3. GBF GDE
4. BGF DGE
−−−
(Diagonals of a par5. BG GD
allelogram bisect
each other.)
6. BFG DEG
(ASA ASA)
−− −−
7. FG GE
8. G is the midpoint
(Definition of
−−
midpoint)
of FE.
−−− −−
14 1. MX QS
−− −−−
2. TX QM
3. QMXT is a parallelogram.
−−−
4. M is the midpoint of QR.
−−−
5. QM MR
−−
(Opposite sides of
6. QM TX
a parallelogram are
congruent.)
−−− −−
7. MR TX
8. XTS MXT
9. MXT MQT
10. MQT RMX
11. XTS RMX
12. XST RXM
(Transitive
postulate)
(Alternate
interior angles)
(Opposite
angles)
(Corresponding
angles)
(Transitive
postulate)
(Corresponding
angles are
congruent)
(AAS AAS)
13. MRX TXS
15 1. Parallelogram ABCD
−−
2. X is the midpoint of BC.
1 BC
3. XC _
(Definition of a
2
midpoint)
−−−
4. Y is the midpoint of AD.
1 AD
5. AY _
2
−−− −−
6. AD BC
7. AD BC
1 BC
1 AD _
8. _
2
2
9. AY XC
−− −−
10. AY XC
11. AMY CMX
(Vertical angles
are congruent.)
12. MAY MCX
(Alternate interior angles are
congruent.)
13. MAY MCX
(AAS AAS)
−−− −−−
(CPCTC)
14. XM YM
15. a M is the midpoint
(Definition of
−−
midpoint)
of XY.
−−− −−−
(CPCTC)
16. AM MC
17. b M is the midpoint
(Definition of
−−
midpoint)
of AC.
−−
16 1. AC is a diagonal in parallelogram ABCD.
−− −−
2. AF CE
−− −−
3. FE FE
−− −− −− −−
4. AF FE CE FE
−− −−
5. AE CF
6. DAC BCD
(Definition of a
parallelogram)
7. BAF DCE
(Alternate interior angles are
congruent.)
8. EAD FBC
9.
10.
11.
12.
17
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
18 1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
12.
13.
14.
15.
16.
17.
19 1.
2.
3.
4.
5.
6.
(Subtraction
postulate)
−−− −−
AD BC
(Opposite sides of a
parallelogram are
congruent.)
ADE BCF (SAS SAS)
AED CFB (CPCTC)
−− −−
DE BF
(Alternate interior
angles are congruent.)
ABCD is a parallelogram.
−− −−
DE AF
−− −−
CF AF
DAE and CFB are right angles.
DAE CFB
−− −−−
CA AD
−−−−
AEBF
−− −−
EB EB
−− −−
AE BF
(Subtraction
postulate)
DEA CFB (SAS SAS)
−− −−
DE CF
(CPCTC)
Parallelogram ABCD
−−
H is the midpoint of AB.
1 AB
AH _
2
−−−
F is the midpoint of DC.
1 DC
FC _
2
−− −−−
AB DC
(Opposite sides of a
parallelogram are
congruent.)
AB DC
1 DC
1 AB _
_
2
2
AH FC
−−− −−
AH FC
−−− −− −− −−
HG AC, FE AC
HGA and FEC are right angles.
HFA FEC
−− −−−
AB DC
CAB ACD
GAH ECF (AAS AAS)
−−− −−
HG FE
Parallelogram ABCD
−− −−−
AR CM
−− −−− −−− −−−
AR MR CM MR
−−− −−
AM CR
−−− −−
AD BC
−−− −−
AD BC
10-2 The Parralallogram
57
20
7.
8.
9.
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
12.
13.
14.
15.
21
1.
2.
3.
4.
5.
6.
7.
22
1.
2.
3.
4.
5.
6.
7.
58
MAD RCB
MAD RCB
(SAS SAS)
−− −−−
BR DM
(CPCTC)
Parallelogram ABCD
−−−
QD bisects D.
1 mCDP
mCDQ _
2
−−
PB bisects B.
1 mABQ
mABP _
2
CDP ABQ
mCDP mABQ
1 mCDP _
1 mABQ
_
2
2
mCDQ mABP
CDQ ABP
DCQ BAP
ABP CDQ
(SAS SAS)
−− −−−
(CPCTC)
a AP CQ
−− −−−
BC AD
−− −−
(Subtraction
b BQ PD
postulate)
Parallelogram PQRS
PS PQ
mx mQSP
−−− −−
QR PS
y QSP
(Alternate
interior angles
are congruent.)
my mQSP
mx > my
(Substitution
postulate)
Parallelogram MARC
AR MA
mAMR mARM
ARM CMR
(Alternate
interior angles
are congruent.)
mARM mCMR
mAMR mCMR (Substitution
postulate)
AMR is not congruent to CMR.
Chapter 10: Quadrilaterals
10-3 Proving That a
Quadrilateral Is a
Parallelogram
(pages 224–225)
1 Check students’ answers. The following is
one possible solution.
−−−
−−
Slope of AB slope of CD 3. Slope of
−−
−−−
AD slope of BC 0. Both pairs of opposite
sides are parallel.
2 a PR 5
7 , 8 or (3.5, 8)
b _
2
3 a Check students’ answers. The following is
one possible solution.
−−−
DR AB 10. Slope of DR slope
−−
of AB 0. One pair of opposite sides are
both congruent and parallel.
−−−
b The length of the altitude from B to DR
is 5.
Note: Since there are many variations of proofs,
the following is simply one set of acceptable
statements to complete each proof. Depending
on the textbook used, the wording and format
of reasons may differ, so they have not been
supplied for the method of congruence applied
in each problem. (These solutions are intended
to be used as a guide—other possible solutions
may vary.)
4 1. Parallelogram LOVE
2. AOV BEL
3. OAV EBL
−−− −−
4. OV EL
5. OAV EBL
(AAS AAS)
1
5 BF _CE (segment connecting midpoints of
2
1 CE (definsides of a triangle) and CD _
2
1 AC and
tion of midpoint). BF CD. DF _
2
1 AC. BF CD and DF BC. Opposite
BC _
2
sides of a parallelogram have equal measure
so they are congruent.
−−
−−
6 1. KL and EU bisect each other at M.
−−− −−−
2. KM LM
−−− −−−
3. EM MU
4. EMK UML
5. EMK UML
(SAS SAS)
−− −−
(CPCTC)
6. EK LU
−−
7. K is the midpoint of JE.
( )
1 EJ
8. EK _
2
−−−
9. L is the midpoint of UN.
1 UN
10. UL _
2
11. EK UL
1 EJ _
1 UN
12. _
2
2
13. EJ UN
−− −−−
14. EJ UN
−− −−
15. EU EU
16. LUM KEM
17. EUN UEJ
−− −−
18. EN UJ
19. JUNE is a
(Both pairs
parallelogram.
of opposite sides
are congruent.)
7 1. ABCD is a parallelogram.
−−
2. RC
−−− −− −− −−
3. DQ RC, AR RC
4. DQC and ARB are right angles.
5. DQC ARB
6. RBA QCD
(Corresponding
angles are
congruent.)
−− −−−
(Opposite sides of
7. AB DC
a parallelogram are
congruent.)
8. AB DC
9. ARB DQC
(AAS AAS)
−− −−−
(CPCTC)
10. AR DQ
11. AR DQ
−− −−−
Segments perpen12. AR DQ
dicular to the same
segment are
parallel.)
13. ARQD is a
(One pair of opparallelogram.
posite sides is both
congruent and
parallel.)
−−
−−
8 1. PE bisects HL at M.
−−− −−−
2. HM ML
3. EPL PEH
4. HME LMP
5. HME LMP
(AAS AAS)
−− −−
6. HE LP
−− −−
7. HE LP
8. HELP is a
(One pair of opparallelogram.
posite sides is both
congruent and
parallel.)
9
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
12.
13.
10
1.
2.
3.
4.
5.
6.
7.
8.
9.
11
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
12
1.
2.
3.
4.
Parallelogram ABCD
−−− −−
AD BC
−−− −−−
AM NC
−−−
M is midpoint of AD.
1 AD
AM _
2
−−
N is the midpoint of BC.
1 BC
NC _
2
−−− −−
AD BC
AD BC
1 BC
1 AD _
_
2
2
AM NC
−−− −−−
AM NC
ANCM is a
(One pair of opparallelogram.
posite sides is both
congruent and
parallel.)
BR
and DM
2 3
−− −−−
BC AD
1 4
BAD DCB
(Supplements of
congruent angles
are congruent.)
−− −−
BD BD
BCD DAB
(AAS AAS)
−− −−−
BC AD
ABCD is a
(One pair of opparallelogram.
posite sides is both
congruent and
parallel.)
−−
−−
QS bisects PR (at M).
−−− −−−
QM MS
1 2
QMR SMP
QMR SMP
(AAS AAS)
−−− −−
QR SP
−− −−
PR PR
PSR RQP
(SAS SAS)
−− −−
QP SP
PQRS is a
(One pair of opparallelogram.
posite sides is both
congruent and
parallel.)
−−−
−−
QV bisects RT.
−− −−
RS ST
−−− −−
QR PV
RQS TVS
(Alternate interior
angles)
10-3 Proving That a Quadrilateral Is a Paralallogram 59
5.
6.
7.
8.
9.
10.
11.
RSQ TSV
QRS VTS
−−− −−
QR PT
−− −−
PT TV
−−− −−
QR PT
−−− −−
QR PT
PQRT is a
parallelogram.
(Vertical angles)
(AAS AAS)
(CPCTC)
(One pair of opposite sides is both
congruent and
parallel.)
13 1. Parallelogram ABCD
−−− −−
(Opposite sides of
2. DC AB
a parallelogram are
congruent.)
−− −−
3. DF BE
−−−−
4. DFEB
5. CDF ABE
(Alternate interior
angles are congruent.)
6. DFC BEA
(SAS SAS)
−− −−
(CPCTC)
7. CF AE
−−− −−
(Opposite sides of
8. AD CB
a parallelogram are
congruent.)
9. ADF EBA
10. AFD CEB
(SAS SAS)
−− −−
(CPCTC)
11. AF CE
12. AECF is a
(Both pairs
parallelogram.
of opposite sides are
congruent.)
−−
14 1. KJ is a diagonal in parallelogram KBJD.
−− −−
2. KA JC
3. BJC AKD
−− −−−
4. BJ KD
5. BJC KAD
(SAS SAS)
−− −−−
(CPCTC)
6. BC AD
−− −−
7. BK JD
8. CJD BKA
9. ABK CDJ
(SAS SAS)
−− −−−
(CPCTC)
10. AB CD
11. ABCD is a
(Both pairs
parallelogram.
of opposite sides are
congruent.)
−−
15 1. BD is a diagonal in parallelogram
ABCD.
2. BEC AFD
3. EBC ADF
(Alternate interior
angles are congruent.)
−−
(Opposites sides of
4. BC AD
a parallelogram are
congruent.)
60
Chapter 10: Quadrilaterals
5.
6.
7.
8.
9.
10.
11.
CBE AFD
−− −−
AF EC
−− −−
DF EB
ABE FDC
−− −−−
AB DC
ABE CDE
−− −−
EA CF
(AAS AAS)
(CPCTC)
(CPCTC)
(SAS SAS)
(CPCTC)
10-4 Rectangles
(pages 227–228)
1 (1) are congruent
2
AR DR
3(4x 3) 10x 1
x5
AR CR DR BR 51
3
AR BR
2(x 6) 3x 20
x8
AR CR 4
BD 8
4
DR CR
4(3x 10) 3(x 2) 12
46
x_
9
46
AR _
9
128
AC BD _
9
5
AC BD
1
2 (12x 3) 5x
3(2x 5) _(4x 4) _
3
4
x2
AC 24
DR 12
6 2x 30 36
x3
7
PR QS
4x 3 6x 7
x5
PR 4(5) 3 23
QS 6(5) 7 23
PR QS √
23 23 23
√
8 mADB mDAC 90 49 41
9 (5, 6)
Note: Since there are many variations of proofs,
the following is simply one set of acceptable
statements to complete each proof. Depending
on the textbook used, the wording and format
of reasons may differ, so they have not been
supplied for the method of congruence applied
in each problem. (These solutions are intended
to be used as a guide—other possible solutions
may vary.)
10 1. Rectangle ABCD
2. ABCD is a parallelogram.
−− −−−
3. BC AD
−− −−−
(Opposite sides of
4. AB CD
a parallelogram are
congruent.)
−− −−
(Diagonals of a
5. BD CA
rectangle are
congruent.)
6. CDA BAD
(SSS SSS)
7. CAD BDA
(CPCTC)
11 1. Rectangle PQRS
2. PQRS is a parallelogram.
−− −−
3. PQ PQ
−−− −−
4. QR PS
5. SQP PQR
(SSS SSS)
6. 1 2
(CPCTC)
12 1. Rectangle ABCD
2. ABP and NCD are right angles.
3. ABP and NCD are right triangles.
−− −−−
4. AP DN
−−− −−
(Opposite sides of
5. DC AB
a parallelogram are
congruent.)
6. ABP DCN
(HL HL)
7. DNC APB
(CPCTC)
8. PAD APB
(Alternate interior
angles are
congruent.)
9. NDA APB
10. PAD NDA
(Transitive
postulate)
−− −−
(Definition of an
11. AE DE
isosceles triangle)
−−
13 By the addition postulate of inequality, AY
−−
is not congruent to TX, and AMY is not
−−−
congruent to THX. Therefore, MY is not
−−−
congruent to HX.
14 1. Rectangle ABCD
−−−
2. N is the midpoint of CD.
3. CN DN
−−− −−−
4. CN DN
5. BCN NDA
(A rectangle is
equiangular.)
−− −−−
(Opposite sides of
6. BC AD
a parallelogram are
congruent.)
−− −−−
(CPCTC)
7. BN AN
−− −−−
1. AB CD
2. BJH and AJH are linear angles.
3. mBJH mAJH 180
1 mAJH 90
1 mBJH _
4. _
2
2
(Division postulate)
−−
5. JG bisects BJH.
1 mBJH
6. mHJG _
2
−−
7. EJ bisects AJH.
1 mAJH
8. mEJH _
2
9. mHJG mEJH 90 (Substitution)
10. mEJG 90
11. EJG is a right angle.
12. CHJ BJH
13. EHJ GJH
14. AJH DHJ
15. HJE JHG
−− −−
16. HJ HJ
17. EJH GHJ
−− −−−
18. EJ GH
−− −−
19. EH JG
20. EJGH is a parallelogram.
21. EJGH is a rectangle. (Definition of a
rectangle)
16 1. Rectangle PQRS
−− −−
2. PA CS
−− −− −− −−
3. PA AC CS AC
−− −−
4. PC AS
−− −−
5. QP RS
6. QPS RSP
7. QPC RSA
(SAS SAS)
8. a 1 2
(CPCTC)
9. PQR and SRQ are right angles.
10. 3 is complementary to 1.
11. 4 is complementary to 2.
12. b 3 4
(Complements of
congruent angles
are congruent.)
−− −−
(Definition of an
13. c QB RB
isosceles triangle)
15
10-5 Rhombuses
(pages 229–231)
1 mB mD 105; mC 75.
AB CD 7
2 mADB 66
10-5 Rhombuses
61
3 4x 2 3x 3
x5
RS 18
4 Perimeter of MABC 8
5 mADC 110
6 2x 2 x 8
x 10
CD 18
7 mADC 94
−−
−−
8 a Midpoint of AC midpoint of BD (6, 5)
−−
−−
b Slope of AC 1, slope of BD 1.
Slopes are negative reciprocals, diagonals
are perpendicular.
9 a (9, 5)
2
b PQ PS 5 √
−−
−− _
1
c Slope of PR , slope of QS 3.
3
Slopes are negative reciprocals, diagonals
are perpendicular.
−− k
−−−
10 a Slope of AD slope of BC _
x.
AD BC √
x 2 k 2 . ABCD is a parallelogram. (One pair of opposite sides have
the same length and are parallel.)
−−
−− _
k , slope of BD
k ,
b Slope of CA _
xk
xk
not negative reciprocals. The diagonals are
not perpendicular.
Note: Since there are many variations of proofs,
the following is simply one set of acceptable
statements to complete each proof. Depending
on the textbook used, the wording and format
of reasons may differ, so they have not been
supplied for the method of congruence applied
in each problem. (These solutions are intended
to be used as a guide—other possible solutions
may vary.)
−−− −− −− −− −−− −−
11 QR AC, PS AC, QR PS. Likewise,
−− −− −− −−
−− −−
QP BD, RS BD, and QP RS. Thus, PQRS
−− −− −−−
is a parallelogram. Since AC BD, QR
−−
−−
−−
and PS are perpendicular to QP and RS.
Thus, PQRS has at least one right angle, and
PQRS is a rectangle.
12 1. Rhombus ABCD
−−
2. ED
−− −−
3. BF AC
−− −−
4. EF AC
−−
5. F bisects AC.
6. EA EC
62
Chapter 10: Quadrilaterals
−− −−
7. EA EC
8. ACE is isosceles.
13
14
15
16
17
18
(Definition of an
isosceles triangle)
−− −−
−− −−
Prove that DF AB and DE BF by using
the midpoints and alternate interior angles.
Thus, EBFD is a parallelogram. By the divi−− −−
sion postulate, EB FB. EBFD is a rhombus
because a rhombus is a parallelogram with
two congruent consecutive sides.
1. Parallelogram ABCD
−− −−−
2. AB CD
3. BAD 2
(Corresponding angles are congruent.)
4. mBAD m2
5. BAD 1 CAD
6. mBAD m1
(Whole is greater
than a part.)
7. m2 m1
(Substitution
postulate)
1. Rhombus PQRS
−− −−
2. QS PR
−−− −−
3. QC CS
−−−
4. B is the midpoint of QC.
−−
5. D is the midpoint of CS.
−− −−−
6. BC
___ CD
___
7. AC AC
8. ABC ADC
(SAS SAS)
−− −−−
(CPCTC)
9. AB AD
10. BAD is isosceles. (Definition of an
isosceles triangle)
If a quadrilateral is equilateral, it is a parallelogram with a pair of consecutive congruent sides.
The diagonal creates two congruent triangles
(ASA ASA) and corresponding congruent
sides are consecutive.
−−−
1. CD BE
−−− −−
2. CD BA
−−−
3. AD BF
−−− −−
4. AD BC
5. ABCD is a
(Definition of a
parallelogram.
parallelogram)
6. BAD BCD
(Opposite angles
are congruent.)
bisects FBE.
7. BG
8. CBD ABD
−−
9. ABCD is a rhombus. (Diagonal BD
bisects opposite
angles.)
10-6 Squares
(pages 232–233)
1 (4) A rectangle is a square.
2 (3) sides and angles are congruent
3 (4) A trapezoid is a parallelogram.
−− −−−
4 (3) AC DC
5 (1) congruent and bisect the angles to which
they are drawn
2
6 (2) x √
−−−
−− 3
7 Slope of DA slope of VE _. Slope of
4
−−
−−
4 . A parallelogram
ED slope of AV _
3
has two pairs of opposite sides that are parallel. A rectangle is a parallelogram in which
consecutive sides have slopes that are nega−−−
1 . Slope
tive reciprocals. Slope of DV _
7
−−
of AE 7. Slopes of the diagonals are negative reciprocals, thus they are perpendicular.
Therefore, DAVE is a square.
−−−
−−
4 . Slope of
8 Slope of MA slope of TH _
3
−− 3
−−−
HM slope of AT _. A parallelogram has
4
two pairs of opposite sides that are parallel.
A rectangle is a parallelogram in which consecutive sides have slopes that are negative
−−−
1 . Slope of
reciprocals. Slope of MT _
7
−−−
AH 7. Slopes of the diagonals are negative reciprocals, thus they are perpendicular.
Therefore, MATH is a square.
9 The diagonals bisect the vertex angles, creating four congruent isosceles triangles. The
bisected angles (the base angles of the triangles) measure 45. The vertex angles thus
measure 90. The diagonals cross, forming
90 angles.
−−
−−
−−−
−−
10 a Slope of PQ slope of RS 0. QR and SP
have no slope. A parallelogram has two
pairs of opposite sides that are parallel.
A rectangle is a parallelogram in which
consecutive sides are perpendicular.
b PQ RS 20. QR PS 7. Since all
sides are not congruent to each other,
PQRS is not a square.
Note: Since there are many variations of proofs,
the following is simply one set of acceptable
statements to complete each proof. Depending
on the textbook used, the wording and format
of reasons may differ, so they have not been
supplied for the method of congruence applied
in each problem. (These solutions are intended
to be used as a guide—other possible solutions
may vary.)
11 a Given ABCD is a square. BFE is a right
−− −−
angle because EF BD. mEBF 45
−−
because diagonal BD bisects right angle
−− −−
ABC. mFEB 45. Therefore, BF EF.
b 1. ABCD is a square.
−− −−
2. EF BD
3. EFD is a right angle.
4. DAE is a right
(Definition of
angle.
a square)
5. DAE and DFE are right triangles.
−− −− −−
6. (Draw DE). DE DE
7. DAE DFE
(HL HL)
−− −−
(CPCTC)
8. EF EA
−−
12 1. Rhombus ABCD with diagonals AC
−−
and BD intersecting at E.
2. 1 2
−− −−
3. BE CE
−−
−−
4. E bisects AC and BD.
−− −−
5. BD AC
6. ABCD is a rectangle.
7. ABCD has all right angles.
8. ABCD is a square.
(A square is a
rhombus in which
all angles are right
angles.)
10-7 Trapezoids
(pages 236–238)
1 (2) They are congruent.
2 (3) ADC ABC
3 a x z 70; y 110
b x 73; y z 107
c x 40; y 108; z 32
d x 70; y 44; z 66
e x 130; y 20; z 30
f x 82; y z 41
−−−
−− 1
. These legs
4 Slope of MA slope of TH _
2
−−
3 −−−
are parallel. Slope of AT _. HM has
4
no slope. These legs are not parallel.
AT HM 10. Therefore, MATH is an
isosceles trapezoid.
10-7 Trapezoids
63
Note: Since there are many variations of proofs,
the following is simply one set of acceptable
statements to complete each proof. Depending
on the textbook used, the wording and format
of reasons may differ, so they have not been
supplied for the method of congruence applied
in each problem. (These solutions are intended
to be used as a guide—other possible solutions
may vary.)
5 1. ABCD is an isosceles trapezoid.
2. BAD CDA
−− −−−
3. BC AD
4. NBC BAD
(Corresponding
angles are
congruent.)
5. NCB CDA
6. NBC NCB
(Substitution
postulate)
7. NBC is isosceles. (Base angles of an
isosceles triangle
are congruent.)
6 1. Isosceles trapezoid ABCD
−− −−−
2. AB DC
3. BAD CDA
(Base angles of an
isosceles trapezoid
are congruent.)
−−− −−−
4. AD AD
5. ADB DAC
(SAS SAS)
7 1. Trapezoid ABCE
−− −−−
2. BD AD
−− −−
3. AB CE
4. A B
5. CED A
6. ECD B
(Corresponding
angles are
congruent.)
7. CED ECD
(Substitution
postulate)
−−− −−
8. CD ED
−− −−−
(Subtraction
9. BD CD −−− −−
AD ED
postulate)
−− −−
10. BC AE
11. ABCE is an
(Nonparallel sides
isosceles trapezoid. of an isosceles
trapezoid are
congruent.)
8 1. Quadrilateral PQRS
−−− −−−
−−−
2. QAB, RAS, and PSB
−−
−−
3. QB bisects RS.
−− −−
4. RA AS
64
Chapter 10: Quadrilaterals
−−− −−−
5. PSB QR
6. RAQ BAS
7. QRA BSA
8.
9.
9 1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
10
11
12
1.
2.
3.
4.
5.
6.
7.
8.
9.
1.
2.
3.
4.
5.
6.
7.
8.
9.
1.
2.
3.
4.
5.
(Vertical angles are
congruent.)
(Alternate interior
angles are
congruent.)
(ASA ASA)
QRA BSA
−−− −−
QA AB
Isosceles trapezoid ABCD
A D
−−− −−
AD BC
−− −−−
AB DC
−−−
E is the midpoint of AD.
−− −−
AE ED
ABE DCE
(SAS SAS)
−− −−
(CPCTC)
a BE CE
BE CE
BCE is isosceles.
(Definition of isosceles triangle)
−− −−
(The median from
b EH BC
the vertex angle of
an isosceles triangle is perpendicular to the base.)
Isosceles trapezoid PQRS
−− −−
PQ RS
−−− −−
QR PS
P S
−− −−
PS PS
PQS SRP
(SAS SAS)
PQS SRP
(CPCTC)
QAP BAS
(Vertical angles are
congruent.)
PAQ SAR
(AAS AAS)
Trapezoid ABCD
−− −−−
BR AD
−−− −−−
CM AD
ARB and DMC are right angles.
ARB and DMC are right triangles.
−− −−−
AB CD
BAR CDM
ARB DMC
(Hypotenuse–acute
angle)
1 2
(CPCTC)
Trapezoid PQRS
Q R
−− −−− −− −−−
PA QR, SE QR
PAQ and SER are right angles.
PAQ and SER are right triangles.
6. AP ES
−− −−
7. AP ES
8. PAQ SER
−− −−
9. QP RS
(Distances between
parallel lines are
equal.)
(Leg–Acute Angle)
(CPCTC)
10-8 Kites
(page 239)
1 A kite is a quadrilateral with only two pairs
of adjacent congruent sides. A rhombus has
four congruent sides.
2 True
3 True
4 False
5 True
6 True
7 False
8 False
9 True
−−
10 a AC
b 1 8; 2 7; 3 6; 4 5
11 a 90
b 45
c 17
d m4 25 and mKLM 50
Note: Since there are many variations of proofs,
the following is simply one set of acceptable
statements to complete each proof. Depending
on the textbook used, the wording and format
of reasons may differ, so they have not been
supplied for the method of congruence applied
in each problem. (These solutions are intended
to be used as a guide—other possible solutions
may vary.)
−−
−−
12 1. QS is the perpendicular bisector of PR.
−− −−
2. PT TR
3. QTP QTR
−− −−
4. QT QT
5. QPT QRT
(SAS SAS)
6. STP STR
−− −−
7. TS TS
8. STP STR
(SAS SAS)
−− −−−
(CPCTC)
9. QP QR
−− −−
(CPCTC)
10. SP SR
−− −−
11. QT ST
12. PQRS is not
(Definition of
a rhombus.
a rhombus)
13. PQRS is a kite.
(Definition of a
kite)
10-9 Areas of Polygons
(pages 243–245)
1 4
2 16
3 9, 12, 15
4 Length is 4 ft and width is 5 ft.
5 a h 10 x
b A x(10 x)
2
b 9x 2
c
6 a x
2
2
d x 4x 4 e 4x 4x 1
25
7 a _
b 18
c
2
d 1
e 9
8 11
9 a 12
b 45
c
45
3
e _
d 18 √
2
√
√
10 36 144 6 5
11 a BD 10
b 120
√
3
12 72 13 s 6
14 a 13
b 120
c
x 2 4x 4
49
_
2
20
120
_
13
c 4x 12
15 a 21x
b 10x 2
2
d 18x 12
e 2x 6x
16 a x 4
b 4(4) 4 12
17 8
3
18 27 √
√
2
19 30 20 10 and 20
21 a 96
b 260
c 60
22 72
23 100
24 a 6
b SMP and AML are similar. Let x be the
−−
perpendicular drawn from M to AP.
20
x _
Then _
, and x 5.
6x
4
10
x
_
b x4
25 a _
20
12 x
c Area of RCS 20. Area of QCT 80.
4(8 11)
9
26 a A _ 38 b 5, _
2
2
−−
−− _
3
_
27 a Slope of JE , slope of EN 4 . The
3
4
slopes of the two legs are negative reciprocals, therefore perpendicular, forming a
right angle.
1 (JE)(EN) 25
b A_
2
( )
10-9 Areas of Polygons
65
28 Enclose PAT in a large rectangle and subtract the excess areas.
a Area of PAT 120 60 24 6 12
18
b AT 10
c 3.6
52 ; sides are congruent.
29 a SA SM √
−− _
−−−
3
2;
Slope of SA , slope of SM _
2
3
slopes are negative reciprocals, therefore
perpendicular and forming a right angle.
1 (SA)(SM) 26
b A_
2
c √
26
30 Subdivide pentagon SIMON into two triangles and a trapezoid and take the sum of all
the areas.
Area 72
31 Divide pentagon JANET into two triangles
and take the sum of the two areas.
Area 64
32 Enclose quadrilateral RYAN in a large rectangle and subtract the excess areas.
13
Area 72 _ 6 12 12 35.5
2
−−
−−
33 NI and CK have zero slope, the two sides are
−−− −−
10 . Therefore, NICK
parallel. NK IC m √
is an isosceles trapezoid.
−−
−−− 1
; the sides are
34 Slope of JO slope of HN _
3
−−
parallel. Slope of NJ 3. The slopes are
negative reciprocals, therefore perpendicu−−−
lar, forming a right angle. Slope of OH is not
−−
equal to the slope of NJ, therefore, only one
pair of sides are parallel. JOHN is a right
trapezoid.
35 a 38
b 28
c 28
d (5, 4)
Chapter Review (pages 245–248)
1 (4) The diagonals of a parallelogram bisect
each other.
2 (4) mB mC 360
3 (1) rhombus
4 (1) a rhombus
5 (3) A rhombus is a square.
6 (4) parallelogram
7 (3) 3
−−
−−
8 (4) QS and PR bisect each other.
9 (1) the diagonals are congruent
10 a square
b rhombus
c parallelogram
d rectangle
66
Chapter 10: Quadrilaterals
11 3x 40 x 50
x5
12 5
13 5x 90
x 18
14 (3x 20) (7x 40) 180
x 20
15 2
16 mDAE 90 75 15
17 (4, 3)
18 3x 15 7x 55
x 10
19 a mACD mCAB 30
b rectangle
180 75
20 mAEK _ 52.5
2
21 5 ft
22 (1, 1)
2
23 3 √
24 mA 45
25 h 8
26 Rhombus. Since the triangles are congruent,
the opposite angles that are originally vertex
angles are congruent. By the addition postulate, the opposite angles formed by joining
the triangles at their bases are congruent.
Thus the quadrilateral is a parallelogram.
All sides are congruent. The parallelogram is
a rhombus.
27
PS QR
2x 3 x 2
x5
PS QR SR PQ 7
28 a (2x 8) 3(x 34) 180
x 14
mABC mCDA 144
mDAB mBCD 36
b AE EC
4y 6y 36
y 18
BE ED
3x 1 x 13
x7
AC 144, BD 40
29 mABP mBAP 90
(5x 10) (2x 4) 90
x 12
mDCB 140, mAPB 90
30 a mR 150
b mRAD 150
c mGAD 135 d mD 30
20 √
3 35
20
200 √3 346.4
2 . ABC is isosceles beAB BC 5 √
cause there is one pair of congruent sides.
−−
−−
b Slope of AB 1. Slope of BC 1. Slopes
−− −−
are negative reciprocals, thus AB BC.
c Area of ABC 25
−− −−
−−
33 DE AV because the slope of DE slope
−− 1 −−−
−−
of AV _
. DA is not parallel to VE because
3 −−−
the slope of DA is not equal to the slope
−−
of VE. DA VE 5; nonparallel sides are
congruent. Therefore, DAVE is an isosceles
trapezoid.
−−
−− 3
34 Slope of PQ slope of RS _. Slope of
4
−−
−−−
−−
4 . Slope of PR
1.
PS slope of RQ _
_
3
7
−−
Slope of QS 7. PQRS is a parallelogram
because opposite sides have the same slope,
and a rhombus because the diagonals are
perpendicular—slopes are negative reciprocals, and a square because consecutive sides
are perpendicular—slopes are negative reciprocals.
−−−
−−
35 Slope of AB slope of CD 0. Slope of
−−
−−
−−−
4 . Slope of AC
BC slope of AD _
2.
3
−−
1 . ABCD is a parallelogram
Slope of BD _
2
because opposite sides have the same slope,
and a rhombus because the diagonals are
perpendicular—slopes are negative
reciprocals.
−−
−−
4 . Slope
36 Slope of LE slope of AF _
5
−−
−−
−− 4
of EA slope of FL _
. Slope of LA is
−− 5
udefined. Slope of EF 0. LEAF is a parallelogram because opposite sides have the same
slope, and a rhombus because the diagonals
are perpendicular.
Note: Since there are many variations of proofs,
the following is simply one set of acceptable
statements to complete each proof. Depending
on the textbook used, the wording and format
of reasons may differ, so they have not been
supplied for the method of congruence applied
in each problem. (These solutions are intended
to be used as a guide—other possible solutions
may vary.)
31 a
b
c
32 a
37
1.
2.
3.
4.
5.
6.
7.
8.
Quadrilateral TRIP
−− −−
TR RI
m1 m2
1 2
−− −−
RA RA
a RAT RAI
−− −−
TA AI
TAP IAP
(SAS SAS)
(CPCTC)
(Supplements of
congruent angles
are congruent.)
−− −−
AP AP
TAP IAP
(SAS SAS)
b 3 4
(CPCTC)
Parallelogram PQRS
−−
Diagonal QS bisects PQR.
PQS RQS
RQS QSP (Alternate interior
angles are congruent.)
5. PQS QSR (Alternate interior
angles are congruent.)
6. RQS QSR (Transitive postulate)
7. QSP QSR
−− −−
8. PS PQ
9. PS PQ
10. PQRS is a rhombus. (A rhombus is a
parallelogram with
two congruent
consecutive sides.)
39 1. ABED is a rhombus.
−−−−
−−
2. BD intersects AOEC at O.
3. BOE and DOE are right angles.
4. BOE and DOE are right triangles.
−− −−
5. OE OE
−− −−
6. BE DE
7. BOE DOE
(HL HL)
−− −−−
(CPCTC)
8. BO OD
9. BOC and DOC are right triangles.
−−− −−−
10. OC OC
11. BOC DOC
(Leg-leg)
−− −−−
(CPCTC)
12. a BC DC
−− −−−
13. AB AD
−− −−
14. AC AC
15. ABC ADC
(SSS SSS)
16. b ABC ADC (CPCTC)
40 Mark the point of intersection of the diagonals E. The diagonals bisect each other,
thus AE DE. By the triangle inequality
theorem, mADE mDCE and
mADC mDCB by the multiplication
postulate of inequality.
9.
10.
11.
38 1.
2.
3.
4.
Chapter Review
67
CHAPTER
Geometry of Three
Dimensions
11-1 Points, Lines, and
Planes
(pages 251–252)
1 False
2 False
3 True
4 True
5 True
6 True
7 True
8 False
9 True
10 False
11 False
12 H
13 E
14 C
15 G
16 C
17 G
18 E
19 D
−− −− −− −−− −− −−
20 a AB FG, AB HC, AB ED,
−−− −−− −−− −− −−− −− −− −−
AH DG, AH CB, AH EF, FA GB,
−− −−− −− −−
FA DC, FA EH
b There are many possible answers. The following are just a few.
−− −− −− −−− −− −− −−
AB & EF, AB & DG, BC & EF, BC &
−− −−− −−− −−− −−
ED, DC & AH, DC & EF.
21 Infinitely many
22 Infinitely many
23 Infinitely many
24 One
25 One
26 Infinitely many
68
Chapter 11: Geometry of Three Dimensions
11
27
28
29
30
31
32
33
34
35
Infinitely many
One
Infinitely many
Infinitely many
One
One
One
None
One
11-2 Perpendicular Lines,
Planes, and Dihedral
Angles
(pages 257–258)
1 False
2 True
3 True
4 False
5 True
6 True
7 False
8 True
9 One
10 Infinitely many
11 Four
12 Infinitely many
13 One
14 One
15 a ABCD
b GFE
16 a BCDE and GCDE
b PRS, SRT, ADE, and EDF
17 The plane that bisects the dihedral angle
−−− −− −−− −−− −− −−
18 QR, QS, QU, WP, TP, VP
Note: Since there are many variations of proofs,
the following is simply one set of acceptable
statements to complete each proof. Depending
on the textbook used, the wording and format
of reasons may differ, so they have not been
supplied for the method of congruence applied
in each problem. (These solutions are intended
to be used as a guide—other possible solutions
may vary.)
19 1.
2.
3.
4.
5.
6.
7.
plane M
BC
BCA and BCD are right angles.
BCA BCD
−− −−
AB DB
−− −−
BC BC
BCA BCD
(HL HL)
BAC BDC
(CPCTC)
11-3 Parallel Lines and
Planes
(pages 259–260)
1 Parallel
2 Perpendicular
3 Perpendicular
4 Parallel
5 Parallel
6 Parallel
7 Parallel
8 Infinitely many
9 None
10 One
11 Line a is parallel to plane M. Line b is contained in plane R. Line b is in plane M. Lines
a and b are coplanar and do not intersect.
Therefore, a b.
12 True
13 True
14 True
15 False
16 False
17 True
18 False
19 False
20 True
21 True
22 True
23 True
24 True
25
26
27
28
29
30
True
False
False
False
False
True
11-4 Surface Area of
a Prism
(pages 264–265)
1 (3) 80
2 a 104
b 108
c 224
3 4 faces, 7 edges, 4 vertices
4 5 faces
5 Right prism
2
2
b 223 ft
6 a 404 cm
7 208
8 242 square inches
9 588
2
10 435 in.
2
11 Lateral area: 255 cm ; total area:
25 √
3
255 _ cm 2
2
12 Lateral area: 45 cm 2; total area:
9 √
3
45 _ cm 2
2
13 144 32 √
3 cm 2
14 3,880 in. 2
3
15 Lateral area: 72; total area: 72 12 √
16 Lateral area: 510 square inches; total area:
3 square inches
510 75 √
17 3 √3 in.
2
2
2
18 d 2 h 2 (AB) , but (AB) l 2 w . By sub2
stitution, d 2 h 2 l 2 w
11-5 Symmetry Planes
(pages 267–268)
1 (1) A
2 (2) E
3 (1) F
Exercises 4–7: Check students’ sketches.
8 7 symmetry planes
9 Check students’ sketches.
10 112.5 √3
11-5 Symmetry Planes
69
11-6 Volume of a Prism
(pages 269–270)
1 288
3
2 15,625 m
3
2
3 a Volume: 105 cm ; surface area: 142 m
3
2
b Volume: 3,600 in. ; surface area: 1,500 in.
3
2
c Volume: 48 ft ; surface area: 88 ft
4 147
5 a 2
b 6
c 4
d 5
e 3
6 14
1
7 _
2
8 Volume: 1,920 ft 3; surface area: 992 ft 2
9 1
3
3 ft
10 Volume: 2,197 ft ; diagonal: 13 √
11 30 inches
3
12 480 cm
13 9
14 2.7 ft
3
15 x √
√
3
16 81 17 Volume: 343; surface area: 294
3 in. 3
18 24 √
19 Volume is ten times as large.
3 cubic
20 a Each solid has a volume of 360 √
units.
b triangular prism 360 square units,
6
hexagonal prism 120 √
2
2
2
21 Diagonal, d √l
w h , so d 2 2
l 2 w h 2. Multiply by four, so that
2
2
4d 4l 2 4w 4h 2.
11-7 Cylinders
(pages 273–274)
1 (3) 6 inches
2 h 11
3 V_
9
1
_
4 r
4
1
5 h_
6
7
8
9
10
70
a
a
a
a
a
112 cm 2
2
96 in.
2
24 in.
2
6 m
r3
b
b
b
b
b
136.5 cm
168 in. 2
26 in. 2
14 m 2
r4
2
c
c
c
c
196 cm
288 in. 2
12 in. 2
6 in. 2
3
Chapter 11: Geometry of Three Dimensions
11 A cylinder with a radius of 8 and a height of
12 has a volume of 768. A cylinder with a
radius of 12 and a height of 8 has a volume
of 1,152. Their difference is 384.
12 38 pounds
13 r 3.4
14 h 29.4
15 a 64
b The volume is twice as
large; 128
c 32
d The lateral area stays
the same; 32.
1 of the
16 a Doubled
b Volume is _
4
volume
original.
17 Yes. The volume of the glass with a diameter
of 2.4 is 1.44. Double the volume is 2.88.
The volume of the second glass is 2.89,
which is greater than twice the volume of the
first glass.
18 False
19 True
2
20 Lateral surface area: 960 in. ; volume:
3
3,840 in.
6,930
385
21 h _ _
36
2
22 16 : 64 or : 4
23 a V 256 b V 512
c V 256 512
11-8 Pyramids
(pages 278–279)
2
1 234 cm
2
2 240 cm
2
3 1,152 in.
3
4 4,848 cm
3
5 57.2 in.
3
6 7.5 ft
3
7 48 √
8 h 300 ft
9 h6
2
10 279 ft
3
11 864 in.
12 120
3
b 340
13 a 180 25 √
3
c 360 150 √
14 9 √3
15 Volume of pyramid is 80. Volume of prism
is 240.
16 240
3 cm 2
17 144 √
3 ft 3
18 432 √
11-9 Cones
(pages 283–284)
1 Lateral area: 60; total area: 96;
volume: 96
2 Lateral area: 20; total area: 36;
volume: 16
3 Lateral area: 136; total area: 200; volume:
320
4 Lateral area: 80; total area: 105
2
3
5 Lateral area: 135 cm ; volume: 324 cm
2
2
6 Lateral area: 260 in. ; total area: 360 in. ;
3
volume: 800 in.
7 r3
8 V 36
9 V 16
10 V 12
11 r 10
1 base height
12 _
3
13 equal
14 three times
15 one-third
16 No, because the slant height is always
greater than the radius.
17 4 : 1
18 h 8
3
19 28.5 in.
3,056
20 Volume: _ 1,018.67 in. 3; lateral area:
3
300 in. 2
3
21 144 √
22 60
11-10 Spheres
(pages 287–288)
1 a r 10.5 ft
2 a r 5 mi
c r 4 mi
3
3 a 288 in.
b r 6.2 ft
c r 2.9 ft
√
b r 2 5 4.5 mi
d r 1.6 mi
b 36 ft 3
3
1 yd 3
d _
c 972,000 mi
6
4 m 3
e _
81
4 r 7 in.
500
2
3
5 Surface area: 100 cm ; volume: _ cm
3
6 Surface area: 192 in. 2; volume: 256 √3 in. 3
7 Volume: 1,679,616,000 cubic miles; surface
area: 4,665,600 square miles
8 63,361,600 square miles
9 1,000,000 times
10 r 5
11 r 3
9
12 9 : 25 or _
25
4
2
b 4 : 25 or _
13 a 2 : 5 or _
5
25
2
4
14 a _
b _
3
9
15 a m 2 : n 2
b m3 : n3
64
16 _
27
17 108 ounces
18 1,458 lb.
2
4
b r_
19 a r _
√
√
24
_
20 r 1 ft 3
4 ft 3
21 a _
b _
c 4.5 ft 3
6
3
22 64
23 h 4r
24 : 6 or _
6
2
_
25 2 : 3 or
3
26 They are equal.
2
Lateral area of cylinder 16 in.
2
Surface area of sphere 16 in.
√
Chapter Review (pages 289–290)
2
1 (4) 216 in.
2 False
3 False
4 True
5 True
6 False
7 False
8 False
9 False
10 True
11 True
12 a 8 vertices, 6 faces, 12 edges
b 6 vertices, 5 faces, 9 edges
c 8 vertices, 6 faces, 12 edges
d 10 vertices, 7 faces, 15 edges
13 a Surface area: 216, volume: 432
b Surface area: 90, volume: 100
2
3
c Surface area: 52 ft , volume: 24 ft
Chapter Review
71
14
15
16
17
18
19
20
21
22
23
47.5 in. 2
h 4 in.
3
25 cm
The cylinder formed by spinning the rectan3
gle about the shorter side is 96 cm greater
in volume.
3
r 6 and h 8, V 288 cm
3
r 8 and h 6, V 384 cm
2
3 in.
36 9 √
3 in. 3
36 √
3 cm 3
144 √
480
2
156 in.
√
65 in.
24
25
26
27
28
29
30
31
32
3
100 ft
2
60 ft
2
3
Lateral area: 15 in. ; volume: 12 in.
3
r 3 in.; volume: 36 in.
2
Surface area: 40,000 cm
4,000,000
1 cm 3
Volume: _ cm 3 or 1,333,333_
3
3
2
3
4
_
Surface area: 4 ft ; volume: ft
3
8
r _ meters
√
5
_
inches or 2.5 inches
2
40.57 41 scoops
CHAPTER
Ratios, Proportion,
and Similarity
12-1 Ratio and Proportion
(pages 294–295)
1 No
2 Yes
3 No
4 Yes
3
5 a _
2
_
b 2
5
3
c _
5
5
d _
3
6 77
7 15
8 9
9 15
10 5
72
Chapter 12: Ratios, Proportion, and Similiarity
12
11 44
12 15
13 39
mt
14 _
a
2ma
_
15 r
4ar
16 _
t
17 9
18 8
19 15
20 4 √3
11
21 3 √
22 5 √3
23 6 √2
24 4 √6
1
25 _
10
√
3
1 or _
26 _
9
3 √3
27
28
29
30
31
32
33
34
40, 50
70, 110
20, 35
16, 20, 24
length 28, width 49
45, 135
a 8
b 6
a 20
b 9
23 a 1.
2.
3.
4.
5.
6.
7.
8.
9.
c 10
12-2 Proportions Involving
Line Segments
(pages 298–300)
1 a–d are all true
4 Proportions are not true, lines not
2 _
2 _
5
8
parallel.
3 6
4 28
5 5.4
6 6
7 24
8 5
ac
9 _
b
np
_
10 m
11 6
12 20
c(a b)
13 _
a
14 No
15 No
16 No
17 No
18 Yes
19 12
1
20 1 : 3 or _
3
21 11 cm
22 x 6, AB 47, DE 17, AC 34, EC 9,
BE 9, AD 23.5, DB 23.5
Note: Since there are many variations of proofs,
the following is simply one set of acceptable
statements to complete each proof. Depending
on the textbook used, the wording and format
of reasons may differ, so they have not been
supplied for the method of congruence applied
in each problem. (These solutions are intended
to be used as a guide—other possible solutions
may vary.)
b
24 1.
2.
ABC
−−
P is the midpoint of AB.
−−
Q is the midpoint of BC.
−− −−
PQ AC (A line joining the midpoints
of two sides of a triangle is
parallel to the third side.)
−− ___
PQ AR
−−
R is the midpoint of AC.
−− −−−
AB RQ
−− −−−
AP RQ
PQRA is a parallelogram. (Definition
of a parallelogram)
30
ABC, ADC
−−
−− AC
wx
−−
−− AC
zy
−− zy
−−
wx
BAD, BCD
−−
−− BD
wz
−−
−− BD
xy
−− xy
−−
wz
wxyz is a
parallelogram.
25 48 √3
3.
4.
5.
6.
7.
8.
9.
(A line dividing two
sides of a triangle proportionally is parallel to
the third side.)
(Definition of a
parallelogram)
12-3 Similar Polygons
(pages 302–303)
1 1:1
2 4, 4.5, 5
3 1:4
4 45, 55
5 a 2:3
b 4.5, 7.5
c 18
6 9, 13.5, 18
7 5a, 5b, 5c
8 All corresponding angles are congruent, all
corresponding sides are in proportion, 1 : 2.
y _
x, _
w, _
, z
9 _
3 3 3 3
10 32, 40
11 mR mQ 110
12 mI 40; mK 160
13 12
12-3 Similar Polygons
73
b QR 10, RS 20,
ST 12, PT 22
15 a True
b False
c True
d True
e False
f True
g False
h False
16 If the vertex angles are congruent, then the
base angles must also be congruent. Similarly, if the base angles are congruent, then
the vertex angles are congruent. Triangles are
similar by (AA) or (AAA).
14 a 3 : 2
12-4 Proving Triangles
Similar
(pages 307–308)
1 (2) similar
2 Vertical angles are congruent. (AA)
3 Not similar
Note: Since there are many variations of proofs,
the following is simply one set of acceptable
statements to complete each proof. Depending
on the textbook used, the wording and format
of reasons may differ, so they have not been
supplied for the method of congruence applied
in each problem. (These solutions are intended
to be used as a guide—other possible solutions
may vary.)
−− −−
4 a 1. AE BC
2. BCD DAE
(Alternate interior
angles)
−−
−−
3. AC and BE intersect at D.
4. BDC ADE
(Vertical angles)
5. ADE CDB
(AA)
b AD 30
5 All corresponding sides are in
BE _
AB _
AE _
4 . Triangles
proportion; _
5
DE
CD
CE
are similar.
5
4 and _
6 _
, corresponding sides are not in
6
3
proportion. Triangles are not similar.
6
4 , corresponding sides are not in
7 _ and _
7
5
proportion. Triangles are not similar.
−−− −−
8 1. CD AB
2. CDA and CDB are right angles.
3. CDA CDB
4. CAD CBD
5. CAD CBD
(AA)
74
Chapter 12: Ratios, Proportion, and Similiarity
−− −−
9 1. AB DE
2. EDF BAC
−− −−
BC EF
EFD BCA
ABC DEF
(AA)
−−− −−
−− −−
BE AC and CD AB
HEC and HDB are right angles.
HEC HDB
DHB CHE
(Vertical angles are
congruent.)
5. EHC DHB
(AA)
−− −−
−− −−
1. DE BC and DF AB
2. DEC and DFA are right angles.
3. DEC DFA
4. Parallelogram ABCD
5. FAD ECD
6. AFD CED
(AA)
1. DBE ADF
2. BAC BDE
(Corresponding
angles)
3. B B
4. DBE ABC
(AA)
1. Parallelogram ABCE
−−−
2. AED
−− −−
3. BC AE
−− −−−
4. BC AD
5. CBF ADB
(Alternate interior
angles)
6. BAD FCB
7. BAD FCB
(AA)
1. ABC PQR
2. ABC PQR
−−
3. BD bisects ABC.
−−
4. QS bisects PQR.
5. ABD PQS
6. BAD QPS
7. ABD PQS
(AA)
1. Parallelogram ABCD
−−− −−
2. AD BC
3. GAE BCG
(Alternate interior
angles are
congruent.)
−−
4. AC bisects HAE.
5. HAG GAE
6. HAG BCG
−−− −−
7. AH AE
3.
4.
5.
10 1.
2.
3.
4.
11
12
13
14
15
(Corresponding
angles are
congruent.)
8.
9.
10.
11.
GEA GHA
GEA GBC
GHA GBC
HAG BCG
12-6 Proving Proportional
Relationships Among
Segments Related to
Triangles
12-5 Dilations
(pages 310–311)
80
10
b _
c 9
1 a _
3
3
2 a BC 10, PA 2.2, BA 11, PS 9.6
10
5
b __
8
4
3 (21, 9)
4 (6, 12)
5 (9, 0)
6 (27, 3)
7 (12, 12)
8 (10, 36)
9 (20, 12)
10 (2.5, 1)
11 (4, 2)
12 (3, 0)
13 (3, 3.5)
2 , 2.5)
14 (2 √
15 (16, 8)
16 (10, 2)
17 (15, 0)
18 (3, 15)
19 (3, 2)
20 (4, 8)
21 D 2 r x-axis
22 D 3 r y-axis
23 r x-axis D _1
2
24 r x-axis D _1
4
25 a A(0, 0), B(12, 0), C(15, 6), D(3, 6)
b Slope AB 0; slope BC 2; slope CD 0;
slope DA 2
c Midpoint M (2.5, 1) and midpoint
M (7.5, 3). Yes, M is the image result
of D 3 operating on point M. Midpoints are
preserved under dilation.
(pages 315–316)
1 3 : 4 and 3 : 4
2 5 cm
3 4:9
4 11 in.
5 AD 6, DC 8
6 18
7 15
8 4:7
9 AC 23, PR 8
10 RQ 6, BC 12.2, AC 20.6
11 13
19
12 _ or 4.75
4
13 22
14 12
15 a 4 : 7
b 16 and 20
48
4
c 84 and 48
d __
7
84
16 a BQ 8, QR 5, PR 8
b 36
12-7 Using Similar Triangles to Prove Proportions
or to Prove a Product
(pages 319–321)
1 128
2 40
3 108
Note: Since there are many variations of proofs,
the following is simply one set of acceptable
statements to complete each proof. Depending
on the textbook used, the wording and format
of reasons may differ, so they have not been
supplied for the method of congruence applied
in each problem. (These solutions are intended
to be used as a guide—other possible solutions
may vary.)
−− −−
−− −−
4 a 1. ED AC and AB CB
2. EDA and ABC are right angles.
3. EDA ABC
12-7 Using Similar Triangles to Prove Proportions or to Prove a Product
75
5
6
7
8
76
4. A A
5. ADE ABC
(AA)
AC _
AB
_
b
AE
AD
c 32
a Each smaller triangle is similar to the
larger triangle so they are similar to each
other.
b Corresponding sides of similar triangles
are in proportion.
c Product of means product of extremes.
d BD 6
1. MAH is a right angle.
−− −−−
2. UT AH
3. UTH is a right angle.
4. MAH UTH
5. H H
6. MAH UTH
(AA)
MA
AH
(Corresponding
7. _ _
UT
TH
sides of similar
triangles are in
proportion.)
−− −−
1. AB DE
2. CAB CED
(Alternate interior
angles are
congruent.)
3. ACB ECD
(Vertical angles are
congruent.)
4. ACB ECD
(AA)
AB
ED
_
_
5.
AC
EC
AC
AB
_
_
(Corresponding
6.
ED
EC
sides of similar
triangles are in
proportion.)
1. EBA CDA
2. A A
3. EBA CDA
4. BCF DEF
−− −−
5. AC AE
6. DEC BCE
(Isosceles triangle
theorem)
7. DEC DEF (Subtraction
BCE BCF
postulate)
8. FEC FCE
9. CBD EDB
10. CBD EDB
(AA)
BC
BE
11. _ _
DE
DC
Chapter 12: Ratios, Proportion, and Similiarity
12. BC DC BE DE
9 1. EBC CDE
2. BFC DFE
3. DEF BCF
FB _
FD
4. _
FE
FC
FC
FB
_
_
5.
FE
FD
10 1.
2.
3.
4.
5.
6.
7.
11 1.
2.
3.
4.
5.
6.
7.
12 1.
2.
3.
4.
5.
6.
7.
(The product of the
means equals the
product of the
extremes.)
(Vertical angles are
congruent.)
(AA)
(Corresponding sides
of similar triangles are
in proportion.)
−− −−
AE BC
ADC and EDC are right angles.
ADC EDD
DAC DEC
DEC DAC (AA)
AC
EC
_
_
ED
AD
EC AD (The product of the
AC ED
means equals the
product of the
extremes.)
−−− −−−
Isosceles triangle WXZ with WX WZ
X Z
YTW YRW
XTY ZRY
(Supplements of congruent angles are
congruent.)
XTY ZRY
(AA)
YR
YT _
_
XY
ZY
XY
YT _
_
(Corresponding sides
YR
ZY
of similar triangles are
in proportion.)
Rectangle DEFG
DGB and EFC are right angles.
DGB EFC
Isosceles triangle ADE
BDG CEF
(Supplements of congruent angles are
congruent.)
BDG CEF
(AA)
BG
DG _
_
(Corresponding sides
EF
CF
of similar triangles are
in proportion.)
6
8
2
240 cm
32 in.
5 , 2x 20 √
5
x 10 √
24
5
19.2 in.
Perimeter is 56. Area is 192.
3
10
3 √3
2
2
BD a b 2
2
2
BD d 2 c
2
2
a b2 d2 c
2
2
a d2 b2 c
2
2
2
25 AB BC AC
2
2
2
AC AD CD
2
2
2
2
AB BC AD CD
12
13
14
15
16
17
18
19
20
21
22
23
24
12-8 Proportions in a
Right Triangle
(pages 323–324)
1 (1) 10
21
2 (3) √
3 (3) 4
77
4 (1) √
5 a False
b True
c False
d False
e True
f True
g True
h False
15
6 2 √
7 5
8 6
55
9 √
5 , y 4 √5, z 4
10 x 2 √
5 , y 4 √5, z 8
11 x 2 √
√
12 x 3 6 , y 6 √3, z 6
3 , y 8 √3, z 12
13 x 4 √
6 , z 40 √
6
14 x 20, y 8 √
15 x 9, y 6.75, z 18.75
5 , y 10
16 x 4 √
10 , y 3 √6, z 3 √
15
17 x 3 √
√
√
18 x 12, y 4 5 , z 6 5
3
19 x 9, y 6, z 6 √
20 x 25, y 17.5, z 29.2
21 x 2.4, y 3.2, z 1.8
2
22 a x 21
b (10) x(x 21) c 4
23 13
26 1, LM 4 √
26
24 a LR 2 √
b 9 and 10
Special Triangles
12-9 Pythagorean Theorem
and Special Triangles
(pages 327–329)
1 (2) 10 √2
2 (3) 25 feet
13
3 (1) √
4 (1) 50 miles
41
5 (4) √
6 (1) 24 feet
7 (2) 13
8 (4) {10, 24, 26}
9 14
10 b, c, d, e, f, g
2
11 a 3 √
d 9
b √
3
5
e _
12
c √
5
f
√
14
(pages 332–334)
2 ft
1 (1) 6 √2 ft and 6 √
√
2 (3) 1 : 3 : 2
3 (4) 6 √2
4 (1) 1
5 (2) 6 √3
6 (1) 30
7 (3) 3.75
8 (2) 2 √6
9 (3) 60
10 a BC 7, AC 7 √3
3
b BC 2.5, AC 2.5 √
√
c BC 3 3 , AC 9
d AC 9 √3, AB 18
3
e AC 12, AB 8 √
f BC 7, AB 14
3 , AC 8.25
g BC 2.75 √
h BC 4 √3, AB 8 √3
2
11 a AC 4, AB 4 √
2
b BC 10, AB 10 √
c AC 8, BC 8
d AC 1.5, BC 1.5
e AC 6 √2, BC 6 √2
f BC 3 √2, AB 6
g BC 10 √2, AB 20
2 , BC 7.5 √2
h AC 7.5 √
√
i BC 4 3 , AB 4 √6
j AC 4 √6, BC 4 √6
12-9 Pythagorean Theorem and Special Triangles
77
12
13
14
15
16
17
18
19
20
21
7 √
2
4
3
5 √
3 , y 12, z 6 √
2
x 6 √
a 6 by 6 √
3
b 12 12 √3
3
c 36 √
2
9 √
46
3 , AC 24, DC 8, DB 16
AB 16 √
2.28
a 8
b 2
12-10 Perimeters, Areas,
and Volumes of Similar
Figures (Polygons and
Solids)
(pages 336–337)
1 (1)1 : 3
2 (3) 3 : 5
3 (1) 28
4 a 9:1
b
d 25 : 1
e
5 a 2:3
b
d 2:5
e
6 1:9
7 343
8 13.5
9 25 : 1
2
10 539 ft
11 1 : 25
12 27 : 64
13 48
3
14 27 ft
15 27.7 in.
16 506.25 kg
17 a 1 : 9
18 a Yes
b 2:3
16 : 81
2:3
6:5
9 : 11
c 81 : 4
c 1:7
b 1:3
c 4:9
d 8 : 27
312 _
4
_
e Ratio of surface areas:
702
9
8
360
_
_
Ratio of volumes:
27
1,215
19 72
20 a 2.4
b 27 : 125
78
Chapter 12: Ratios, Proportion, and Similiarity
Chapter Review (pages 338–341)
1 (1) 4 : 25
2 (4) 5 : 14
3 (3) 15
a
4 (2) x _
a
5 (3) √3
6 (4) 64 pounds
an
7 a 6
b 6
c _
m
8 a 5:6
b 2:9
c n:m
d c : (a b)
e p : (m n)
f (m n) : (a b)
g (n b) : (a m)
10
9 a 6
b 8a
c _
3
10 30
11 11.25
12 6 : 35
13 6
14 6
15 14
ac bc
16 _
a
8
5
20
8
_
17
_ , EC _, AC _
3
8x
3
x4
18 a 3
b Expansion
19 54
20 102
3
21 Yes, the ratio of similitude is _.
2
1.
22 Yes, the ratio of similitude is _
2
6
_
23 mQ 125, PQ , QR 2
5
24 2
25 13
13 , BC 6 √
13
26 BD 12, AB 4 √
27 20, 21, 29
b 50
28 a 5 √2
29 34
3 , y 80
30 x 40 √
3, y 8
31 x 8 √
3, y 9
32 x 6 √
2
33 √
34 126
2
35 75 ft
8
36 _
27
37 80 in. 3
38 81
39 45
40 18
2
3 in.
41 72 √
42 Area 60, perimeter 24 10 √2
Note: Since there are many variations of proofs,
the following is simply one set of acceptable
statements to complete each proof. Depending
on the textbook used, the wording and format
of reasons may differ, so they have not been
supplied for the method of congruence applied
in each problem. (These solutions are intended
to be used as a guide—other possible solutions
may vary.)
−− −−
43 1. TS QP
2. QRP TSR
(Corresponding
angles are
congruent.)
3. PQR STR
4. R R
5. PQR STR
(AAA AAA)
−− −− −− −−
44 1. AE BC, BD AC
2. FEB and BDC are right angles.
3. FEB BDC
4. FBE FBE
5. BFE BCD
(AA)
−− −−−
45 1. AB CD
2. ABE DCE
(Alternate interior
angles are
congruent.)
3. BAE CDE
4. AEB CED
5. a ABE DCE
EC
BE _
6. _
ED
AE
BE _
AE
7. b _
ED
EC
8. c BE ED EC AE
(Vertical angles are
congruent.)
(AAA AAA)
(Corresponding
sides of similar
triangles are in
proportion.)
(The product of
means equals
the product of
extremes.)
−− −−
BC AF
ECA and FDA are right angles.
ECA FDA
A A
ADF ACB
BA = _
AF
6. _
BC DF
7. BA DF (The product of
AF DC
means equals
the product of
extremes.)
46 1.
2.
3.
4.
5.
CHAPTER
Geometry of the Circle
13-1 Arcs and Angles
(pages 345–346)
1 a 36
b 51
d 180
e 2r
2 a BC, CD, AD, BD, AB
, ADC
b ABC
, BDA
, ADB
c ABD
c 90
13
75, AS
105, AR
75
105, BS
RB
a 155
b 25
c 155
d 335
a 296
b 244
c 296
d 244
a 73
b 132
c 155
d 180
e 155
f 228
g 253
h 205
i 287
7 a 20
b 60
c 120
d 110
e 7
f 60
g 70
h 120
i 110
j 230
k 240
l 290
3
4
5
6
13-1 Arcs and Angles
79
13-2 Arcs and Chords
(pages 350–351)
m
1 a 16
b 7.5
c _
2
6
e 5 √
2
d 3 √
2 a True
b True
c False
3 109
4 a 90
b 72
c 60
d 36
5 a 24
b 22.5
c 20
d 18
e 15
2
c 5
6 a 10
b 5 √
2
c 12
d 6 √2
7 a 6
b 6 √
8 6 in.
9 17
10 a 15
b 7.5
11 6.
2
12 8 √
2
13 x √
14 5
15 25
16 a 30
b 60
c 60
d 60
e 60
f 300
g 150
17 a 30
b 60
c 10
3
d 5
e 5 √
Note: Since there are many variations of proofs,
the following is simply one set of acceptable
statements to complete each proof. Depending
on the textbook used, the wording and format
of reasons may differ, so they have not been
supplied for the method of congruence applied
in each problem. (These solutions are intended
to be used as a guide—other possible solutions
may vary.)
−−−
−−−
18 1. Radii OK and OM
2. AOL DOL
(Central angles are
congruent.)
−− −−− −− −−−
3. LA OK, LD OM
4. LAO and LDO are right angles.
5. LAO and LDO are right triangles.
6. LAO LDO
−−
−−
7. LO LO
8. LAO LDO
(Leg-angle)
19 1. ABD CBD
−−− −−
(All radii of the
2. OA OB
same circle are
congruent.)
3. BAO ABD
−−− −−
4. OC OB
5. BCO CBD
80
Chapter 13: Geometry of the Circle
6.
7.
8.
20 1.
2.
3.
4.
AOB COB
AOD COD
CD
AD
(Congruent inscribed
angles intercept congruent arcs.)
−−− −−
−−−
OA, OB, and OC are radii of circle O.
BAC BCA
BC
AB
(Congruent inscribed
angles intercept congruent arcs.)
AOB COB
(Central angles are
equal to their arcs.)
13-3 Inscribed Angles
and Their Measure
(pages 354–357)
1 a 25
b 55
c 125
d 60.4
e 4
2 a 52
b 124
c 15
d 180
e (16x)
3 mx 70, my 35; 70 35 105
4 a 50
b 130
c 56
d 25
e 62
f 25
5 53
6 70
7 80
8 35
9 140
10 mC 75, mM 82
11 85.5
12 m1 120, m2 45, m3 75, m4 60
13 m1 57, m2 33, m3 90, m4 114
14 m1 74, m2 74, m3 37, m4 106
15 a 140
b 80
c 40
d 70
e 70
16 m1 57, m2 48, m3 57, m4 48
17 a 80
b 188
c 46
d 94
18 a 64
b 108
c 140
d 70
e 24
f 54
g 94
19 a 86
b 102
c 88
d 56
e 116
20 a 44
b 88
c 44
d 44
e 92
f 88
21 a 60
b 60
c 60
d 30
e 60
f 120
g 30
h 60
i 120
j 60
k 90
l 30
22 Label arcs with degree measures that sum
to 360, as with x, 2x, 3x, 4x. Then find the
values of the inscribed angles.
23 Parallel lines cut congruent arcs x and y.
Therefore,
2x 2y 360
x y 180
Each inscribed angle is equal to one-half the
intercepted arc: 90.
13-4 Tangents and Secants
(pages 363–365)
1 a
b
c
d
2 a 12
b 3
3 a 3.25
b 1
4 a disjoint
b externally tangent
c intersecting twice
d internally tangent
e disjoint internally
f concentric
5 a Sketch of two circles externally disjoint
b Sketch of two intersecting circles, not
tangent
c Sketch of externally tangent circles
d Sketch of internally tangent circles
6 a 1
b 3
c 2
d 4
7 a 160
b 130
c 90
d 40
e 180 x
8 a 60
b 45
c 35
d 51
e 67.5
180 n
n
b 90 2n
c _
9 a _
2
2
180 m n
d 45 n
e __
2
10 16
11 80
12 48
13 234
14 a 18
b 72
c 29.25
15 a 11, 17, 18
b no sides equal
16 a 12, 16, 20
b Satisfies Pythagorean theorem:
2
2
2
12 16 20
17 a 10, 15, 15
b two sides are equal
18 BE 4, EC 5, CF 5, AF 6, AC 11,
AB 10
19 AB 20, CB 20, RB 10
20 AB 24, DR 14, DB 32
21 OB 26, DB 36, RB 16
91 , CB 2 √
91
22 OB 20, AB 2 √
23 OB 22, CB 8 √6, AB 8 √6
24 AB 20, CB 20, OC 15, OB 25
25 Tangent segments from the same point are
congruent. Base angles are equal, each measuring 60. Therefore, APB is equilateral.
Note: Since there are many variations of proofs,
the following is simply one set of acceptable
statements to complete each proof. Depending
on the textbook used, the wording and format
of reasons may differ, so they have not been
supplied for the method of congruence applied
in each problem. (These solutions are intended
to be used as a guide—other possible solutions
may vary.)
−−
26 1. Let E be the point of intersection of AB
−−−
and CD.
−−
−−
2. EB and ED are tangent segments to
circle P.
−− −−
(Two tangent segments
3. EB ED
drawn from an external
point are congruent.)
−−
−−
4. EA and EC are tangent segments to
circle O.
−− −−
5. EA EC
−− −−
(Addition postulate)
6. EA EB −− −−−
EC CD
−− −−−
or AB CD
13-4 Tangents and Secants
81
−−−
−−−
Common external tangents, AG and CM
−−−
−−−
Radii OA and OC
OAG and OCM are right angles.
OAG and OCM are right triangles.
−−− −−−
OA OC
−−−
−−−
Tangents OM and OG of circle P
−−− −−−
OM OG
(Two tangent segments drawn from an
external point are
congruent.)
8. OAG OCM (HL HL)
−−− −−−
(CPCTC)
9. AG CM
28 1. Circles A and B are congruent with
.
tangent FG
−−
−−
2. AF is the radius of A. BG is the radius of
B.
−− −−
3. AF BG
4. AFC and BGC are right angles.
5. FCA GCB
6. FCA GCB
(AAS AAS)
−− −−
7. AC BC
1 (CD
BE
). 2(mA) 29 mA _
2
mCOD mBOE. But mBOE mA.
2(mA) mCOD mA. Therefore,
3(mA) mCOD.
−−−
−−
30 1. AB is tangent to O at E; CD is tangent to
O at F.
−−
2. OE OF
−−− −−− −− −−−
3. OA OC; OB OD
−− −−
4. OE AB
5. OEA and OEB are right angles.
−− −−−
6. OF CD
7. OFC and OFD are right angles.
8. OEA OFC
(HL HL)
OEB OFD
−− −−
(Addition
9. AE EB −− −−
CF FD
postulate)
−− −−−
or AB CD
27 1.
2.
3.
4.
5.
6.
7.
13-5 Angles Formed by
Tangents, Chords, and
Secants
(pages 368–371)
1 (2) 96
2 (1) 24
3 (2) 144
82
Chapter 13: Geometry of the Circle
(4) 173
(1) 38
(1) 90
a 40
b 62
c 225
d 45
e 75
f 100
g 80
h 30
8 a 79
b 38
c x 50, y 60, z 120
9 a 66
b 80
c 73
d 130
e 33
f 73
10 a 90
b 120
c 90
d 60
e 30
f 60
g 30
h 90
11 a 120
b 40
c 80
d 40
e 140
f 100
CD
because in a regular hexagon,
12 BC
congruent chords subtend congruent arcs.
Chords and a tangent that intercept con−−
BD
gruent arcs are parallel. Therefore, PC
Note: Since there are many variations of proofs,
the following is simply one set of acceptable
statements to complete each proof. Depending
on the textbook used, the wording and format
of reasons may differ, so they have not been
supplied for the method of congruence applied
in each problem. (These solutions are intended
to be used as a guide—other possible solutions
may vary.)
−− −−
13 1. FB EC
2. BEC EBF (Alternate interior angles are congruent.)
3. EHJ BHG
4. EHJ BHG
BH _
HE
(Corresponding sides
5. _
HJ
HG
of similar triangles are
in proportion.)
6. BH JH (The product of means
HE HG
equals the product of
extremes.)
−− −−−
14 1. BA DA
−−−
2. Diameter DC
3. BAD and DBC are right angles.
−−
4. AB is tangent to O at B.
1 mBD
5. mDBA _
2
1 mBD
6. mDCB _
2
7. mDBA mDCB
8. DBA DCB
9. DBA DCB
4
5
6
7
DC
BD _
10. _
DB
DA
BD _
DA
11. _
BD
DC
2
12. (BD) DC DA
(Corresponding sides of
similar triangles are in
proportion.)
1
(The product of means
equals the product of
extremes.)
13-6 Measures of Tangent
Segments, Chords and
Secant Segments
(pages 374–376)
1 (4) 20
2 (2) 46
3 (1) 17
4 (3) 24
5 (2) 10
6 (1) 3
7 (3) 25
cz
8 (3) _
a
9 (3) 17
10 (3) 33
11 (1) 9
12 24
13 19
14 40
15 44
16 12
17 16
18 NQ 6, QP 16
19 x 4, GH 16
−−− −−−
2
2
20 AOB WZ . Radius 12.5; 10 (7.5) 2
(12.5)
2
3
4
13-7 Circle Proofs
(pages 381–382)
Note: Since there are many variations of proofs,
the following is simply one set of acceptable
statements to complete each proof. Depending
on the textbook used, the wording and format
of reasons may differ, so they have not been
supplied for the method of congruence applied
in each problem. (These solutions are intended
to be used as a guide—other possible solutions
may vary.)
5
;
1. A is the midpoint of CD
−−−
AOB is a diameter.
AD
2. CA
1 mCA
1 mAD
_
3. _
2
2
1 mAD
4. mABC _
2
1 mCA
mABC _
2
5. mABC mABD
6. ACB and ADB are right angles.
7. mACB mADB
−− −−
8. AB AB
9. ACB ADB
(AAS AAS)
−− −−
(CPCTC)
10. CB DB
−−− −−
1. AD CB
CB
2. AD
1 mAD
3. mABD _
2
1 mCB
mCDB _
2
4. mABD mCDB
−− −−
5. DB DB
1 mDB
6. mDAB _
2
1 mDB
mDCB _
2
7. mDAB mDCB
8. ADB CBD
(AAS AAS)
−− −−−
(CPCTC)
9. AB CD
−−−−
−−−−
1. Circle O with diameters MOT and AOH
2. MAT and HTA are right angles.
−−−− −−−−
3. MOT AOH
−− −−
4. AT AT
5. MAT HTA
(HL HL)
−−− −−
(CPCTC)
6. MA HT
1. Circle O, tangents PR
, PV
; PV
2. PR
OR
OV
3. ORP and OVP are right angles.
4. mORP mOVP
−−− −−−
5. OR OV
−− −−
6. PR PV
7. RPO VPO
(SAS SAS)
8. RPO VPO
(CPCTC)
1. Circle O with T as the midpoint of CH
2. mCT mTH
180
3. mCTH
90
4. mCT
5. mCOT mCT
6. COT and CTH are right angles.
7. mCOT mCTH
13-7 Circle Proofs
83
1 mTH
8. mHCT _
2
1 mCT
mCHT _
2
9. mHCT mCHT
10. CTO HTC
CT
TO
11. _ _
(Corresponding
TH
CH
sides of similar
triangles are in
proportion.)
−−−− GI
6 1. Circle O with diameter DOG; DR
1
_
)
TG
2. mTEG m(DR
2
1 m(GI
TG
)
3. mTRI _
2
GI
TGI
4. TG
1
5. mTEG _mTGI
2
1 mTGI
6. mTRI _
2
7. mTEG mTRI
8. mRTI mRTI
9. TEX TRI
(AA)
EX
TX
10. _ _
TI
RI
11. TX RI EX TI (The product of
means equals
the product of
extremes.)
7 1. CT CH
−−− −−
2. HA TD
3. CTD CHA; TDH HAT
4. TDH is supplementary to TDC;
HAT is supplementary to HAC.
5. TDC HAC
6. TDC HAC
(AAS AAS)
−− −−−
(CPCTC)
7. CT CH
8. HA TD
(Contradiction)
−− −−−
8 1. BC QR
−− −−
2. BA QP
RQ
(Congruent chords
3. BC
subtend congruent
arcs.)
4. BA QP
5. A D
(Inscribed angles
intercepting congruent arcs are
congruent.)
6. B Q
7. RQP CBA
(SAS SAS)
8. AC PR
84
Chapter 13: Geometry of the Circle
9 a 1. BEA CED
2. BDC CAB
(Vertical angles are
congruent.)
(Congruent
inscribed angles)
(AA)
3. BEA CED
16
b (i) 4 : 5 (ii) _
25
−− −−−
10 1. BA PGD
2. DBA BDP
1 mBGD
3. mPBD _
2
1 mBGD
4. mBAD _
2
5. mPBD mDAB
6. PBD DAB
7. PBD DAB
BD
PD _
(Corresponding
8. _
BD
BA
sides of similar
triangles are in
proportion.)
−−
11 1. AB is the diameter.
2. Line l is tangent to circle O.
3. ABC and AEB are right angles.
4. ABC AEB
5. CAB BAE
6. ABC AEB
(AA)
AC
AB
(Corresponding
7. _ _
AB
AE
sides of similar
triangles are in
proportion.)
−−
12 1. AC is the diameter.
−−
2. BD intersects the diameter at point E.
−− −−
3. AB BE
4. BAE BEA
5. BEA CED
6. BAE CED
7. ABE and DCE are right angles.
8. ABE DCE
9. ABE ECD
(AA)
AC _
ED
_
10.
AB
EC
11. AC EC (The product of
ED AB
means equals
the product of
extremes.)
13 1. DCG CBD
1 m(ABC
DF
)
2. mE _
2
1 m(CB
)
_
2
1 m(CB
)
3. mCDB _
2
4. mE mCDB
5. CBD FCE
EC
EF _
6. _
DB
CD
14 1. AZT MZH
2. HMT TAH
3. MZH AZT
MZ
AZ
4. _ _
ZH
ZT
15 a 1. BAC BDC
2. ABD ACD
3. ACD ABE
b BE 6
(AA)
7
(Corresponding
sides of similar
triangles are in
proportion.)
(AA)
8
(Corresponding
sides of similar
triangles are in
proportion.)
(Congruent inscribed angles)
10
(AA)
11
9
12
13-8 Circles in the
Coordinate Plane
13
(pages 386–387)
2
2
1 (2) (x 4) (y 3) 36
2 (4) (13, 1)
2
2
3 a (x 1) (y 4) 25
2
2
b (x 3) (y 2) 49
2
2
c (x 4) (y 1) 121
2
2
d (x 2) (y 5) 64
2
2
e (x 2) y 121
2
2
f x y 144
2
2
4 a x y 49
2
2
b (x 4) y 25
2
2
c x y 41
2
2
d (x 2) y 36
2
2
e (x 3) (y 2) 9
2
2
f (x 1) (y 3) 25
5 For parts a–d, check students’ graphs. The
center and radius of each circle are listed
below.
a C(2, 4), r 3
b C(3, 4), r 5
c C(0, 2), r 4
d C(6, 0), r 2
6 a C(0, 0), r 9
b C(0, 0), r 11
c C(0, 0), r 20
d C(0, 0), r 1
e C(0, 0), r √2
Note: Students’ answers may vary for the
two other points on the circle.
a C(5, 3), r 8; (5, 11), (5, 5)
b C(4, 0), r 10; (6, 0), (14, 0)
c C(2, 5), r 4; (2, 5), (6, 5)
d C(3, 7), r 2 √3; (3, 7 2 √3),
(3, 7 2 √3)
2
2
a x y 25
2
2
b x y 4
2
2
c x y 34
2
2
d x y 37
2
2
e x y 68
2
2
a (x 2) (y 3) 2
2
2
b (x 3) (y 7) 80
2
2
a x (y 0.5) 12.25
2
2
b (x 3) (y 3) 5
a Circumference 8, area 16
b Circumference 12, area 36
13 , area 13
c Circumference 2 √
a (3, 0)
2
2
b (x 3) (y 1) 1
c a C(1, 3), r 2 √2
b Area 8, circumference 4 √2
13-9 Tangents, Secants,
and the Circle in the
Coordinate Plane
(pages 392–393)
1 (1) 0
2 a (3, 11), (3, 1)
b (3) x 1
3 y0
4 y 0.5x 2.5
5 yx4
6 y 3
7 y4
8 y 2x 10
9 a (4, 3), (3, 4)
10 a (0, 4), (4, 0)
11 a (2, 3), (2, 5)
12 a (0, 3)
13 a (0, 2), (2, 0)
14 a (5, 5)
15 a (1, 4), (1, 4)
b
b
b
b
b
b
b
secant
secant
secant
tangent
secant
tangent
secant
13-9 Tangents, Secants, and the Circle in the Coordinate Plane
85
16
17
18
19
20
21
22
a
a
a
a
a
a
a
b
23 a
b
c
24 a
d
(2 √
2 , 2 √
2 ), (2 √
2 , 2 √
2 ) b secant
(10, 0) and (10, 0)
b secant
(7, 7) and (7, 7)
b secant
(6, 0)
b tangent
(8, 6) and (6, 4)
b secant
(0, 5) and (4, 3)
b secant
2
Sub-in the given points in (x 3) 2
(y 2) 25.
−−−
Midpoint of ME is (3.5, 2.5);
0.5 .
distance √
y x 8
yx8
(0, 8)
y 3x 10 b x 10
c P(10, 20)
√
√
√
PA 160 4 10 , PB 1000 10 , PM 20
10 √
2
PA PB PM
10 10 √
10 20 2
4 √
400 400
Chapter Review (page 393–397)
1 (1) All chords in a circle are congruent.
2
2
2 (2) (x 4) (y 2) 9
3 (2) 1
4 (3) 8
5 (3) 6 and 15
6 (2) 12
7 (1) 12
8 (2) 6
9 (2) 68
10 (3) 122
11 (1) 43
12 (3) 86
13 (2) 2 : 1
14 (1) 41
15 (3) 10 inches
16 (3) 125
17 (1) 40
18 (4) 100
19 (2) 230
20 16
21 90
22 104
23 20
24 48
25 12
26 a 40
b 40
c 40
d 40
e 110
f 110
g 35
86
Chapter 13: Geometry of the Circle
40
b 70
c 20
45
e 25
f 65
30
b 45
c 15
135
e 90
C(0, 0), r 20
Answers may vary. (0, 20), (0, 20)
C(7, 11), r 9
Answers may vary. (7, 20), (7, 2)
C(3, 13), r 6
Answers may vary. (3, 19), (3, 7)
5
C(3, 5), r 2 √
5 ),
Answers may vary. (3, 5 2 √
√
(3, 5 2 5 )
2
2
33 (x 3) (y 2) 16, C(3, 2), r 4
2
2
34 a (x 4) (y 2) 49
2
2
b (x 5) y 2
2
2
c x y 169
2
2
d C(3, 0), r 5, (x 3) y 25
2
10 , (x 4) (y 4) 2 10
e C(4, 4) r √
2
2
2
35 6 6 (6 √2)
1 x 10
36 y _
3
37 a (4, 3) and (4, 3)
b secant
38 a (0, 3) and (3, 0)
b secant
39 a (8, 2) and (2, 8)
b secant
40 a (0, 5) and (3, 4)
b secant
Note: Since there are many variations of proofs,
the following is simply one set of acceptable
statements to complete each proof. Depending
on the textbook used, the wording and format
of reasons may differ, so they have not been
supplied for the method of congruence applied
in each problem. (These solutions are intended
to be used as a guide—other possible solutions
may vary.)
41 1. AOB COB
−− −−
2. AB CB
CB
3. AB
−−−
42 1. DA is the diameter of circle O.
−−− −−−
−−−
2. Radii OA, OC, and OD
−−− −−− −−−
(Radii are equal.)
3. OA OC OD
−−− −−−
4. AD CD
5. AOD COD
(SSS SSS)
6. AD CD
7. ABD CBD
(Congruent arcs
are intercepted
by congruent inscribed angles.)
27 a
d
28 a
d
29 a
b
30 a
b
31 a
b
32 a
b
43 1.
2.
3.
4.
5.
6.
7.
−−− −−− −−− −−
MA OK, MD OL
OAM and ODM are right angles.
−−− −−−
AM DM
−−− −−−
OM OM
OAM ODM
(HL HL)
AOM LOM
(CPCTC)
MK ML
(Congruent angles
have congruent
arcs.)
−− −−
−−
44 1. OE AB, OD AC
2. AEO and ADO are right angles.
−− −−
3. AB AC
−−
−−
4. OE bisects AB.
−−−
−−
5. OD bisects AC.
−− −−−
6. AE AD
−−− −−−
7. OA OA
8. AEO ADO
(HL HL)
45 Use equal arcs are subtended by equal
chords.
CHAPTER
Locus and
Constructions
14-1 Basic Constructions
(page 404)
Note: For exercises 1–20, check students’ constructions; procedures may vary.
1 ab Use construction of congruent angles
procedure.
2 Use construction of a perpendicular to a line
through a given point on the line procedure.
3 ac Use construction of an angle bisector
procedure.
4 Use construction of a line perpendicular to
a line through a given point not on the line
procedure.
5 Use construction of a perpendicular bisector
of a line segment procedure.
6 Use construction of a parallel line through a
given point not on the line procedure.
7 Use construction of the median of a triangle
procedure.
8 Use construction of line perpendicular to a
line through a given point not on the line
procedure.
14
9 Use construction of a perpendicular bisector
procedure.
10 Use construction of an angle bisector
procedure.
11 a Use construction of a perpendicular line
procedure.
b Use construction of a perpendicular line
procedure and then construct an angle
bisector.
12 a Use construction of an equilateral triangle
procedure.
b Using angle constructed in a, construct an
angle bisector.
c Construct angle bisector of b.
13 Using the vertex of the angle as the center
of a circle and one of the legs as its radius,
construct a circle. Construct the diameter by
extending the radius.
14 Using the vertex of the angle as the center
of a circle and one of the legs as its radius,
construct a circle. Construct the diameter by
extending the radius. Construct a perpendicular through the center of the circle.
14-1 Basic Constructions
87
15 a Use constructing a congruent angle
procedure.
b Use constructing a congruent angle
procedure.
c Construct a perpendicular bisector of a
segment to form a 90 angle. Then use
constructing a congruent angle procedure
to construct the sum.
d Use constructing an angle bisector
procedure.
e Use constructing an angle bisector
procedure for A. Then use constructing a
congruent angle procedure to construct
the sum.
16 Use construction of an equilateral triangle
procedure.
17 Use the construction of an equilateral
triangle procedure using the length of the
longer segment to construct the leg of the
isosceles triangle.
18 Using the compass, measure the radius and
construct a circle passing through the point
on the tangent line.
19 Use construction of a line tangent to a given
circle through a given point outside the circle
procedure.
20 Use procedures for constructing parallel lines
and perpendicular lines.
14-2 Concurrent Lines and
Points of Concurrency
(pages 407–408)
1 (4) obtuse
2 (3) at one of the vertices of the triangle
3 Check students’ constructions.
4 All at the same point
5 10
2
6 6_
3
7 36
8 16.5
9 4.5
10 SP 4, SC 6
11 PB 15, PR 30
12 PA 15, QA 45
1
13 4_
3
14 9
88
Chapter 14: Locus and Constructions
15
16
17
18
19
20
21
22
23
24
25
3 √3
QE 4, DE 6
a, b, c: all interior
a, b, c: all interior
a interior
b on a side
c exterior
a interior
b at a vertex
c exterior
√
√
Incenter: 3 , circumcenter: 2 3
3 , circumcenter: 4 √3
Incenter: 2 √
√
3 , circumcenter: 6 √3
Incenter: 3 Incenter: 6, circumcenter: 12
Since AG GC, the base is the same and the
altitude is the same. Therefore, the area is the
same.
14-4 Six Fundamental Loci
and the Coordinate Plane
(pages 413–414)
1 (3) One concentric circle of radius 11 inches
2 (2) one line
3 (2) a point
4 (1) a circle of radius 4 with center at P
5 (4) a circle
Note: For exercises 6–17 check students’ sketches.
6 Circle with given point as center and
radius 3
7 Two parallel lines, one on each side of the
given line
8 One line parallel to and midway between the
given parallel lines
9 The perpendicular bisector of the segment
joining R and S
10 The perpendicular bisector of the segment
AB
11 A circle with radius 2.5 inches and the given
point as the center
12 The line that is the bisector of ABC
13 Two lines each the bisector of the vertical
angles
14 Two concentric circles with radii 5 and 9
15 One concentric circle midway between the
given circles
16 All the points in the interior of a circle with
the given point as the center and with a
radius 2 inches
17 A circle with the given point as the center
and a radius of 3 inches and all the points in
the exterior of that circle
18 Two lines, one on each side, parallel to the
given line and at a distance r from the given
line
19 One line parallel to the two parallel lines and
midway between them
20 The diameter of the circle that is the perpendicular bisector of the given chord
21 The diameter of the circle that is the perpendicular bisector of the given chord
22 A ray that is the angle bisector
23 A line perpendicular to the point on the
given line
24 The perpendicular bisector of the chord that
joins the two given points
Locus in the Coordinate Plane
(Pages 418–419)
1 (4) y 2 or y 2
2 (1) x 2
3 (2) x 5
4 (2) y 1
2
2
5 (4) (x 1) (y 3) 25
6 (3) y 3 and x 0
7 (4) y x 1 and y x 5
8 y1
9 y 2, y 10
10 y 4
11 y x 3 and y x 3
12 y x 3
13 x 1 and y 0
2
2
14 a (x 1) (y 4) 16
2
2
b x y 36
2
2
c x (y 1) 9
2
2
d (x 1) y 1
2
2
e x (y 2) 6.25
2
2
f (x 1) (y 3) 30.25
15 The showers could be placed anywhere on
the perpendicular bisector of the line segment joining the diving boards, which is
35 ft from each of the diving boards.
14-5 Compound Locus and
the Coordinate Plane
(pages 421–422)
1 (3) a pair of points
2 (4) the empty set
3 (1) 0
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
(3) a pair of points
(3) 2
(2) 1
a 4
b 2
c 0
Find the point of concurrency (circumcenter)
of the perpendicular bisectors of the sides of
the triangle formed.
3
2
Parallel lines 2 inches from the given line,
one on each side
a Circle with radius 3, center R. 1 point
b Circle with radius 6, center R. 2 points
c Circle with radius 7, center R. 3 points
d Circle with radius 9, center R. 3 points
For all parts, check students’ sketches.
a Two intersecting circles, but not tangent.
2 points
b Two tangent circles. 1 point
c Two disjoint circles. 0 points
Sketch circle and two lines; 4 points.
Sketch a line and a circle; 2 points.
Sketch two lines bisecting vertical angles,
and two parallel lines; 4 points.
Sketch two concentric circles and two lines
bisecting vertical angles; 8 points.
The intersection of the line parallel to m and
k and midway between them, and a circle
with A as the center and d as the radius.
1 f (0 points)
Case I
d
_
2
1 f (1 point)
Case II
d_
2
1
Case III
d _ f (2 points)
2
The intersection of the perpendicular
bisector of segment AB and a circle about
center A with radius x.
1 d (0 points)
Case I
Radius x _
2
1 d (1 point)
Case II
Radius x _
2
1 d (2 points)
Case III
Radius x _
2
Compound Loci and the Coordinate Plane
(pages 423–424)
1 (4) 4
2 (3) 2
3 (4) 4
4 a 2
b 1
c 0
14-5 Compound Locus and the Coorinate Plane
89
5 Two horizontal lines y 11 and y 1, and
vertical line x 4. Locus: 2 points.
6 Locus 2 points: (0, 8) and (8, 0)
7 Locus 2 points: (2, 7) and (2, 1)
8 Locus 2 points: (3, 1) and (2, 2)
9 a y2
b x 4
c (4, 2)
2
2
e one
d (x 2) (y 2) 16
10 a Circle with radius, d
b Two lines: x 1 and x 1
c (i) 1 point
(ii) 3 points (iii) 4 points
14-6 Locus of Points
Equidistant From a Point
and a Line
(page 427)
Note: For exercises 1–10, check students’
sketches.
1 b (0, 2), (2, 3), (2, 3)
2
x 2
c y _
4
2 b (2, 1), (2, 1), (6, 1)
x2 _
x _
1
c y _
2
2
8
3 b (2, 0), (2, 2), (6, 2)
x2 _
1x _
1
c y _
8
2
2
4 b (3, 0.5), (0, 1), (6, 1)
2
x x1
c y _
6
5 b (1, 0), (3, 4), (3, 4)
y2
_
c x
1
8
6 b (3, 1.5), (6, 0), (0, 0)
x2 x
c y _
6
7 b (2, 2), (6, 4), (2, 4)
5
x2 _
1x _
c y _
2
2
8
8 b (1, 4), (4, 10), (4, 2)
y 2 2y
7
c x _ _ _
3
3
12
9 b (0, 0), (2, 4), (2, 4)
y2
c x _
8
10 b (3, 1.5), (1, 2), (6, 2)
x2 x 2
c y _
6
90
Chapter 14: Locus and Constructions
14-7 Solving Other LinearQuadratic and QuadraticQuadratic Systems
(page 430)
Note: For exercises 1–20, check students’
sketches.
1 (2, 0), (2, 0)
2 , 0), (2 √2, 0)
2 (2 √
3 (0, 5), (4, 3)
4 (3, 2)
5 (2, 3), (2, 5)
6 (2, 1), (1, 2)
7 (1, 1), (1, 1)
8 (5, 27), (1, 5)
5
9 _, 6 , (2, 5)
3
10 (4, 0), (4, 0)
11 ( √2, 0), ( √2, 0)
12 (3, 2), (6, 1)
2 , √2), (2 √
2 , √2), ( √
2 , 2 √
2 ),
13 (2 √
2 , 2 √2)
( √
14 (0.5, 1.25), (4, 3)
15 (1, 0)
16 (1, 3), (1, 3)
2 , √5), (2 √2, √5), (2 √
2 , √5),
17 (2 √
(2 √2, √5)
18 (2, 1), (2, 1)
19 (3, 0)
20 (2, 4), (2, 0)
( )
Chapter Review (pages 430–433)
1 (2) acute triangles
−−
2 (4) median to side AC
3 (2) AAS
4 (2) two circles
5 (2) X lies on the locus of points equidistant
from R and S.
6 (2) y 2x 3
2
2
7 (3) (x 3) (y 4) 36
8 (1) x 3
9 (3) x 5
10 (1) (0, 0) and (0, 8)
11 (4) (0, 2) and (4, 2)
12 (3) perpendicular bisectors of the sides of the
triangle
13 (3) 4
14 (3) 3
15 (3) 2
16 (4) 4
17 (1) 90 x
18 8
19 x 3, PD 6, RP 12, RD 18
20 x 2, y 4, BE 9, AD 12
Note: For exercises 21–24, check students’
sketches.
21 Two concentric circles. Locus: radius 1 and
radius 7.
2
2
22 Circle, radius 3, x y 9. Two lines,
x 4, x 4. Locus: 0 points
2
2
2
2
23 Circle: x y 4. Circle: x (y 4) 9.
Locus: 1 point, (0, 2)
24 Two lines parallel to line m on opposite sides
and 4 centimeters from m. Circle with center
H and radius 2. Locus: 1 point
2
2
25 (x 4) (y 2) 1
26 y x 4
27 Two lines: y x 2 and y x 2
Note: For exercises 28–37, check students’
sketches.
28 b (2, 0.5), (5, 1), (1, 1)
2
2x _
x _
1
c y _
6
3
6
29 b (5, 7), (1, 3), (3, 1)
y2 _
3y _
_
c x
1
8
8
4
30 b (2, 0), (2, 2), (6, 2)
31
32
33
34
35
36
37
38
x2 _
x _
1
c y _
8
2
2
b (1, 0), (1, 4), (1, 4)
y2
c x _ 1
8
(0, 5), (3, 4), (3, 4)
(0, 5), (3, 8)
(0, 3), (3, 9)
7 , 1 √
7 ), (2 √
7 , 1 √
7)
(2 √
(3, 10), (1, 6)
7 , 3), ( √
7 , 3)
(0, 4), (√
The intersection of the perpendicular bisector of AB, and circle with A as center
1 f (0 points)
Case I x _
2
1 f (1 point)
Case II x _
2
1 f (2 points)
Case III x _
2
39 The intersection of one circle with radius of
2.5, concentric with the given circles, and
two lines parallel to the given line, one on
each side at a distance x from the given line.
Case I x 2.5 (4 points)
Case II x 2.5 (2 points)
Case III x 2.5 (0 points)
40 The intersection of circle with center P
and radius d and circle with center Q and
radius d.
Case I x 2d (2 points)
Case II x 2d (1 point)
Case III x 2d (0 points)
41 The intersection of two lines parallel to line
m, one on each side at a distance d from line
m and the circle with center A and radius r.
Case I r d (0 points)
Case II r d (2 points)
Case III r d (4 points)
42 Locus: A line parallel to the north side and
south side of the courtyard and midway
between them. Since the width of the garden
is 60 ft, every point on this line will be equidistant from the north and south walls and
at least 30 feet from the North Entrance.
43 Check students’ constructions of the orthocenter of an acute triangle.
44 Check students’ constructions of the perpendicular bisector of the three sides of a right
triangle.
45 Check students’ constructions of a circle that
passes through the three vertices of an obtuse triangle.
Chapter Review
91
!33%33-%.4
Cumulative Reviews
Each review has a total of 58 possible points. Use the following table, adapted from the Regents
Examinations, to convert the student’s raw score to a scaled score.
Raw Score
58
57
56
55
54
53
52
51
50
49
48
47
46
45
44
43
42
41
40
39
Scaled Score
100
99
98
97
96
93
92
91
90
89
87
86
85
84
82
80
79
78
77
76
Raw Score
38
37
36
35
34
33
32
31
30
29
28
27
26
25
24
23
22
21
20
Scaled Score
75
74
72
70
68
67
66
65
64
63
61
60
59
58
57
56
54
53
51
Raw Score
19
18
17
16
15
14
13
12
11
10
9
8
7
6
5
4
3
2
1
Scaled Score
49
47
46
45
44
43
42
41
39
37
36
34
32
29
26
21
16
10
5
Chapters 1–2
(pages 434–437)
Part I
Allow a total of 20 credits, 2 credits for each of the following correct answers. Allow credit if the student has
written the correct answer instead of the numeral 1, 2, 3, or 4.
−−
3 (3) p q
1 (4) a → r
2 (2) AB
92
4 (4) 2BC AC
5 (1) If I win a scholarship, then I will play soccer.
2
6 (3) _
3
7 (3) The triangle may be acute.
8 (4) If two angles are not congruent, then they are not right angles.
9 (2) right
10 (1) isosceles
Part II
For each question, use the specific criteria to award a maximum of 2 credits.
11
12
13
14
Score
Explanation
2
AB CD EF, and an appropriate explanation is given.
1
AB CD EF, but no explanation is given.
0
A zero response is completely incorrect, irrelevant, or incoherent or is a correct response
that was obtained by an obviously incorrect procedure.
Score
Explanation
2
3, and an appropriate explanation is given.
1
3, but no explanation is given.
0
A zero response is completely incorrect, irrelevant, or incoherent or is a correct response
that was obtained by an obviously incorrect procedure.
Score
Explanation
2
55, and an appropriate explanation is given.
1
55, but no explanation is given.
0
A zero response is completely incorrect, irrelevant, or incoherent or is a correct response
that was obtained by an obviously incorrect procedure.
Score
Explanation
2
9, and an appropriate explanation is given.
1
9, but no explanation is given.
0
A zero response is completely incorrect, irrelevant, or incoherent or is a correct response
that was obtained by an obviously incorrect procedure.
Chapters 1–2
93
Part III
For each question, use the specific criteria to award a maximum of 4 credits.
15
16
17
94
Score
Explanation
4
a (i) scalene
(ii) right triangle
b (i) isosceles
(ii) acute
c (i) equilateral
(ii) equiangular
d (i) isosceles
(ii) obtuse
e (i) isosceles
(ii) right triangle
f (i) scalene
(ii) obtuse
3
Answered all but two parts correctly.
2
Answered all but three parts correctly.
1
Answered two parts correctly.
0
A zero response is completely incorrect, irrelevant, or incoherent or is a correct response
that was obtained by an obviously incorrect procedure.
Score
Explanation
4
3 : 1, and an appropriate explanation is given.
3
2
AB is given instead.
An appropriate method is shown, but _
BD
An appropriate method is shown, but an incorrect answer is given.
1
3 : 1, but no explanation is given.
0
A zero response is completely incorrect, irrelevant, or incoherent or is a correct response
that was obtained by an obviously incorrect procedure.
Score
Explanation
4
x 77, and an appropriate method is used.
3
An appropriate method is shown, but y is given instead.
2
An appropriate method is shown, but an incorrect answer is given.
1
x 77, but no explanation is given.
0
A zero response is completely incorrect, irrelevant, or incoherent or is a correct response
that was obtained by an obviously incorrect procedure.
Cumulative Reviews
Part IV
For each question, use the specific criteria to award a maximum of 6 credits.
18
Score
6
Explanation
a distributive property
b associative property of addition
c additive identity
d commutative property of multiplication
19
20
5
Student answers all but one part correctly.
3
Student answers only two parts correctly.
1
Student answers only one part correctly.
0
A zero response is completely incorrect, irrelevant, or incoherent or is a correct response
that was obtained by an obviously incorrect procedure.
Score
Explanation
6
34, and an appropriate method is given.
5
An appropriate method is used, but a single arithmetical error is made.
4
An appropriate method is used, but two arithmetical errors are made.
3
An appropriate method is used, but student found m⬔1 or m⬔2.
2
An appropriate method is shown, but multiple computational mistakes are made.
1
34, but no appropriate explanation is given.
0
A zero response is completely incorrect, irrelevant, or incoherent or is a correct response
that was obtained by an obviously incorrect procedure.
Score
6
Explanation
a {6, 7, 8, 9, 10}
b {0, 2, 4, 6, 8, 10}
c {1, 3, 5}
d {0, 1, 2, 3, 4, 5, 7, 9}
5
One part is missing elements from the solution set.
4
Elements are missing from two solution sets.
3
Answered two parts correctly.
2
Answered all parts, but solution sets are missing elements.
1
Only one part is answered correctly.
0
A zero response is completely incorrect, irrelevant, or incoherent or is a correct response
that was obtained by an obviously incorrect procedure.
Chapters 1–2
95
Chapters 1–3
(pages 438–441)
Part I
Allow a total of 20 credits, 2 credits for each of the following correct answers. Allow credit if the student has
written the correct answer instead of the numeral 1, 2, 3, or 4.
1 (3) 80
2 (2) 54
3 (1) Subtraction postulate
4 (4) If Cecilia is not a senior, then she does not take AP Calculus.
5 (4) 22
6 (3) If a triangle is not equilateral, then it is not isosceles.
7 (1) 20
3
1
9 (4) _
10 (2) b c
8 (3) _
2
2
Part II
For each question, use the specific criteria to award a maximum of 2 credits.
11
12
Score
Explanation
2
70, and an appropriate explanation is given.
1
70, but no explanation is given.
0
A zero response is completely incorrect, irrelevant, or incoherent or is a correct response
that was obtained by an obviously incorrect procedure.
Score
2
Explanation
Affirming the conclusion does not affirm the premise.
a The quadrilateral could be rhombus or any other figure.
b x could be 7 or any other positive number less than 7, so that x is not greater than 8.
13
14
96
1
Student answers only one part correctly.
0
A zero response is completely incorrect, irrelevant, or incoherent or is a correct response
that was obtained by an obviously incorrect procedure.
Score
Explanation
2
m⬔r 20, and an appropriate explanation is given.
1
m⬔r 20, but no explanation is given.
0
A zero response is completely incorrect, irrelevant, or incoherent or is a correct response
that was obtained by an obviously incorrect procedure.
Score
Explanation
2
RB 2, and an appropriate explanation is given.
1
RB 2, but no explanation is given.
0
A zero response is completely incorrect, irrelevant, or incoherent or is a correct response
that was obtained by an obviously incorrect procedure.
Cumulative Reviews
Part III
For each question, use the specific criteria to award a maximum of 4 credits.
15
16
Score
Explanation
4
m⬔A 82, m⬔B 82, and m⬔C 16, and an appropriate explanation is given.
3
An appropriate method is used, but a single arithmetical error is made.
2
An appropriate method is used to find x, but the measures of the angles are not given.
1
m⬔A 82, m⬔B 82, and m⬔C 16, but no explanation is given.
0
A zero response is completely incorrect, irrelevant, or incoherent or is a correct response
that was obtained by an obviously incorrect procedure.
Score
4
Explanation
a ⬔ABD and ⬔DBE
b ⬔ABE and ⬔EBC
c ⬔DBC
d ⬔ABC
17
3
Student answers all but one part correctly.
2
Student answers only two parts correctly.
1
Student answers only one part correctly.
0
A zero response is completely incorrect, irrelevant, or incoherent or is a correct response
that was obtained by an obviously incorrect procedure.
Score
Explanation
4
m⬔4 50, m⬔5 20, m⬔ABD 160, and an appropriate method is used.
3
An appropriate method is used, but a single arithmetical error is made.
2
m⬔4 50, m⬔5 20, but student did not give the measure of ⬔ABD.
1
m⬔4 50, m⬔5 20, m⬔ABD 160, but no appropriate method is given.
0
A zero response is completely incorrect, irrelevant, or incoherent or is a correct response
that was obtained by an obviously incorrect procedure.
Chapters 1–3
97
Part IV
For each question, use the specific criteria to award a maximum of 6 credits.
18
Score
6
Explanation
a If the sum of the measures of two acute angles is 90, the two angles are
complementary.
b If two acute angles are not complementary, then their sum is not 90.
c If the sum of the measures of two acute angles is not 90, then the two acute angles
are not complementary.
d Two acute angles are complementary if and only if the sum of their measures is 90.
19
20
5
Student answers all but one part correctly.
3
Student answers only two parts correctly.
1
Student answers only one part correctly.
0
A zero response is completely incorrect, irrelevant, or incoherent or is a correct response
that was obtained by an obviously incorrect procedure.
Score
Explanation
1
(2) Reflexive postulate
2
(3) Addition postulate
1
(4) Partition postulate
2
(5) Substitution postulate
0
A zero response is completely incorrect, irrelevant, or incoherent or is a correct response
that was obtained by an obviously incorrect procedure.
Score
Explanation
1
(2) Definition of bisector
1
(3) Definition of midpoint
2
(4) Doubles of equals are equal.
2
(5) Substitution postulate
0
A zero response is completely incorrect, irrelevant, or incoherent or is a correct response
that was obtained by an obviously incorrect procedure.
Chapters 1–4
(pages 442–445)
Part I
Allow a total of 20 credits, 2 credits for each of the following correct answers. Allow credit if the student has
written the correct answer instead of the numeral 1, 2, 3, or 4.
1 (3) 220
5 (2) 120
2 (2) a median
6 (1) ⬔3 and ⬔2 are nonadjacent complementary angles.
3 (4) 120
7 (2) b
4 (3) If the diagonals of a quadrilateral
8 (3) ⬔BAR and ⬔RAI
are not congruent, then the
9 (1) m⬔x m⬔y
quadrilateral is not a rectangle.
10 (3) a b 4
98
Cumulative Reviews
Part II
For each question, use the specific criteria to award a maximum of 2 credits.
11
12
13
14
Score
Explanation
2
a True
1
Student answers only two parts correctly.
0
A zero response is completely incorrect, irrelevant, or incoherent or is a correct response
that was obtained by an obviously incorrect procedure.
Score
b False
c False
d True
Explanation
2
11, and an appropriate explanation is given.
1
11, but no explanation is given.
0
A zero response is completely incorrect, irrelevant, or incoherent or is a correct response
that was obtained by an obviously incorrect procedure.
Score
Explanation
2
68 and 112, and an appropriate explanation is given.
1
68 and 112, but no explanation is given.
0
A zero response is completely incorrect, irrelevant, or incoherent or is a correct response
that was obtained by an obviously incorrect procedure.
Score
Explanation
2
57.5, and an appropriate explanation is given.
1
57.5, but no explanation is given.
0
A zero response is completely incorrect, irrelevant, or incoherent or is a correct response
that was obtained by an obviously incorrect procedure.
Part III
For each question, use the specific criteria to award a maximum of 4 credits.
15
Score
Explanation
4
AC 12, AB 16, BC 16, and an appropriate explanation is given.
3
An appropriate method is used, but a single arithmetical error is made.
2
An appropriate method is used, but multiple arithmetical errors are made.
1
AC 12, AB 16, BC 16, but no explanation is given.
0
A zero response is completely incorrect, irrelevant, or incoherent or is a correct response
that was obtained by an obviously incorrect procedure.
Chapters 1–4
99
16
17
Score
Explanation
4
a 91, b 35, c 54, d 91, and an appropriate explanation is given.
3
An appropriate method is used, but a single arithmetical error is made.
2
An appropriate method is used, but multiple arithmetical errors are made.
1
a 91, b 35, c 54, d 91, but no explanation is given.
0
A zero response is completely incorrect, irrelevant, or incoherent or is a correct response
that was obtained by an obviously incorrect procedure.
Score
Explanation
4
m⬔PTC 50, and an appropriate explanation is given.
3
An appropriate method is used, but a single arithmetical error is made.
2
An appropriate method is used, but multiple arithmetical errors are made.
1
m⬔PTC 50, but no explanation is given.
0
A zero response is completely incorrect, irrelevant, or incoherent or is a correct response
that was obtained by an obviously incorrect procedure.
Part IV
For each question, use the specific criteria to award a maximum of 6 credits.
18
Score
3
−− −− −−− −−
a Given ⬔A ⬔F and AB EF. AD CF and by the reflexive property of
−−− −−−
−−−− −−−
congruence DC DC. By the addition postulate, ADC FCD. Therefore, by SAS,
䉭ABC 䉭DEF.
2
a An appropriate proof with correct conclusions is shown, but one reason is faulty
and/or one statement is missing.
1
a A correct conclusion is reached and a reason is given, but no appropriate method
is shown.
−− −−−
−−−
b Given ⬔D ⬔A. C is the midpoint of AD, then AC DC.
⬔ACB ⬔DCE because vertical angles are congruent. Therefore,
by ASA, 䉭ABC 䉭DEC.
3
100
Explanation
2
b An appropriate proof with correct conclusions is shown, but one reason is faulty
and/or one statement is missing.
1
b A correct conclusion is reached and a reason is given, but no appropriate method
is shown.
0
A zero response is completely incorrect, irrelevant, or incoherent or is a correct response
that was obtained by an obviously incorrect procedure.
Cumulative Reviews
19
Score
6
20
Explanation
The following proof or another appropriate proof is given.
Statements
Reasons
1. ⬔B ⬔D
−− −−−
2. BC DC
1. Given.
2. Given.
3. ⬔BCA ⬔DCE
3. Vertical angles are congruent.
4. 䉭ACB 䉭ECD
4. ASA ASA.
5
An appropriate proof with correct conclusion is shown, but one reason is faulty and/or
one statement is missing.
4
A faulty conclusion or no conclusion is drawn from the statements and reasoning.
3
An appropriate proof with correct conclusion is shown, but two reasons are faulty or
two steps are missing.
2
An appropriate proof with correct conclusion is used, but more than two reasons are
faulty or more than two steps are missing or have errors.
1
䉭ACB 䉭ECD and a reason is given, but no appropriate method is shown.
0
A zero response is completely incorrect, irrelevant, or incoherent or is a correct response
that was obtained by an obviously incorrect procedure.
Score
6
Explanation
The following proof or another appropriate proof is given.
Statements
−−
−−
1. BD is the median to AC.
−−
2. D is the midpoint of AC.
−−− −−−
3. AD DC
−− −−
4. AB BC
−− −−
5. BD BD
6. 䉭I 䉭II
Reasons
1. Given.
2. Definition of a median of an isosceles triangle.
3. Definition of a midpoint.
4. Given.
5. Reflexive Property of Congruence.
6. SSS SSS.
5
An appropriate proof with correct conclusion is shown, but one reason is faulty and/or
one statement is missing.
4
A faulty conclusion or no conclusion is drawn from the statements and reasoning.
3
An appropriate proof with correct conclusion is shown, but three reasons are faulty or
three steps are missing.
2
An appropriate proof with correct conclusion is used, but more than three reasons are
faulty or more than three steps are missing or have errors.
1
䉭I 䉭II and a reason is given, but no appropriate method is shown.
0
A zero response is completely incorrect, irrelevant, or incoherent or is a correct response
that was obtained by an obviously incorrect procedure.
Chapters 1–4
101
Chapters 1–5
(pages 446–449)
Part I
Allow a total of 20 credits, 2 credits for each of the following correct answers. Allow credit if the student has
written the correct____
answer instead of the numeral 1, 2, 3, or 4.
−− −−−
6 (1) isosceles
1 (4) QB, QA, QM
−−−
−−
2 (1) q → p
7 (3) CD bisects AB.
3 (3) 70
8 (2) ⬔3 ⬔4
4 (2)
9 (3) 28
5 (4) ⬔ADB ⬔WDB
10 (3) 78.5
Part II
For each question, use the specific criteria to award a maximum of 2 credits.
11
12
13
14
102
Score
Explanation
2
6, and an appropriate explanation is given.
1
6, but no explanation is given.
0
A zero response is completely incorrect, irrelevant, or incoherent or is a correct response
that was obtained by an obviously incorrect procedure.
Score
Explanation
2
x 8, A 81, B 99, and an appropriate explanation is given.
1
x 8, A 81, B 99, but no explanation is given.
0
A zero response is completely incorrect, irrelevant, or incoherent or is a correct response
that was obtained by an obviously incorrect procedure.
Score
Explanation
2
256, and an appropriate explanation is given.
1
256, but no explanation is given.
0
A zero response is completely incorrect, irrelevant, or incoherent or is a correct response
that was obtained by an obviously incorrect procedure.
Score
Explanation
1
a 45, and an appropriate explanation is given.
1
b 90, and an appropriate explanation is given.
1
(a) 45 and (b) 90, but no explanation is given.
0
A zero response is completely incorrect, irrelevant, or incoherent or is a correct response
that was obtained by an obviously incorrect procedure.
Cumulative Reviews
Part III
For each question, use the specific criteria to award a maximum of 4 credits.
15
Score
2
2
0
16
Score
4
Explanation
a ⬔BCA ⬔ACD
−−− −−
b AD AB
A zero response is completely incorrect, irrelevant, or incoherent or is a correct response
that was obtained by an obviously incorrect procedure.
Explanation
The following proof or another appropriate proof is given.
Statements
−− −− −−− −−−
1. DB ⬜ AC, AD ⬜ DC
17
Reasons
1. Given.
2. ⬔DBA and ⬔ADC are right
angles.
2. Definition of perpendicular lines.
3. ⬔1 is complementary to ⬔x.
3. Given.
4. m⬔1 m⬔x 90
4. Definition of complementary angles.
5. m⬔x m⬔2 m⬔ADC
5. Addition postulate.
6. m⬔x m⬔2 90
6. Substitution postulate.
7. m⬔1 m⬔x m⬔x m⬔2
7. Substitution postulate.
8. m⬔1 m⬔2
8. Subtraction postulate.
9. ⬔1 ⬔2
9. Angles with equal measures are congruent.
3
An appropriate proof with correct conclusion is shown, but one reason is faulty and/or
one statement is missing.
2
An appropriate proof with correct conclusion is used, but multiple reasons are faulty or
multiple steps are missing or have errors.
1
⬔1 ⬔2 and a reason is given, but no appropriate method is shown.
0
A zero response is completely incorrect, irrelevant, or incoherent or is a correct response
that was obtained by an obviously incorrect procedure.
Score
2
1
2
Explanation
−−−
−−
−− −−
a Since CD bisects AB at E, AE BE. Given ⬔1 ⬔2, and because vertical angles are
congruent, ⬔AEC ⬔BED. Hence, by ASA, 䉭ACE 䉭BDE.
a A correct conclusion is reached and reason is given, but no appropriate method
is shown.
−− −−−
b Given ⬔1 ⬔2 and LN ⬜ MO. ⬔LNM and ⬔LNO are right angles. Since right
−− −−
angles are congruent, ⬔LNM ⬔LNO. LN LN by the reflexive property of
congruence. Therefore, by ASA, 䉭LMN 䉭LON.
1
b A correct conclusion is reached and reason is given, but no appropriate method
is shown.
0
A zero response is completely incorrect, irrelevant, or incoherent or is a correct response
that was obtained by an obviously incorrect procedure.
Chapters 1–5
103
Part IV
For each question, use the specific criteria to award a maximum of 6 credits.
18
Score
6
104
Explanation
The following proof or another appropriate proof is given.
Statements
Reasons
1. 䉭PET
−−− −− −−− −−
2. PCB ⬜ ET; TCA ⬜ EP
1. Given.
2. Given.
3. ⬔CBT and ⬔CAP are right
angles.
3. Definition of perpendicular lines.
4. ⬔CBT ⬔CAP
4. Right angles are congruent.
5. ⬔ACP ⬔BCP
−− −−
6. PA TB
5. Vertical angles are congruent.
6. Given.
7. 䉭ACP 䉭BCT
−− −−
8. CP CT
7. AAS AAS.
8. CPCTC.
9. 䉭PCT is isosceles.
9. Definition of isosceles triangle.
5
An appropriate proof with correct conclusion is shown, but one reason is faulty and/or
one statement is missing.
4
A faulty conclusion or no conclusion is drawn from the statements and reasoning.
3
An appropriate proof with correct conclusion is shown, but three reasons are faulty or
three steps are missing.
2
An appropriate proof with correct conclusion is used, but more than three reasons are
faulty or more than three steps are missing or have errors.
1
䉭PCT is isosceles and a reason is given, but no appropriate method is shown.
0
A zero response is completely incorrect, irrelevant, or incoherent or is a correct response
that was obtained by an obviously incorrect procedure.
Cumulative Reviews
19
Score
6
Explanation
The following proof or another appropriate proof is given.
Statements
−− −−
1. AC BD
−− −− −− −−
2. AC BC BD BC
−− −−−
3. AB CD
−− −−− −− −−−
4. GB ⬜ AD; EC ⬜ AD
Reasons
1. Given.
2. Subtraction postulate.
3. Partition postulate.
4. Given.
5. ⬔GBA and ⬔ECD are right angles.
5. Definition of perpendicular lines.
6. ⬔GBA ⬔ECD
6. Right angles are congruent.
7. ⬔AGB ⬔DEC
7. Given.
8. 䉭AGB 䉭DEC
8. AAS AAS.
9. ⬔A ⬔D
−− −−
10. AF DF
9. CPCTC.
10. Converse of isosceles triangle theorem.
5
An appropriate proof with correct conclusion is shown, but one reason is faulty and/or
one statement is missing.
4
A faulty conclusion or no conclusion is drawn from the statements and reasoning.
3
An appropriate proof with correct conclusion is shown, but three reasons are faulty or
three steps are missing.
2
An appropriate proof with correct conclusion is used, but more than three reasons are
faulty or more than three steps are missing or have errors.
−− −−
AF DF and a reason is given, but no appropriate method is shown.
1
0
A zero response is completely incorrect, irrelevant, or incoherent or is a correct response
that was obtained by an obviously incorrect procedure.
Chapters 1–5
105
20
Score
6
Explanation
The following proof or another appropriate proof is given.
Statements
−−
1. BD is the median of 䉭ABC.
−− −−
2. BA BC
3. 䉭ABC is isosceles.
−− −−
4. BD ⬜ AC
−−
−−
5. BD is an altitude to AC.
Reasons
1. Given.
2. Given.
3. Definition of isosceles triangle.
4. The median from the vertex angle of an
isosceles triangle is perpendicular to the base.
5. Definition of an altitude.
5
An appropriate proof with correct conclusion is shown, but one reason is faulty and/or
one statement is missing.
4
A faulty conclusion or no conclusion is drawn from the statements and reasoning.
3
An appropriate proof with correct conclusion is shown, but two reasons are faulty or
two steps are missing.
2
An appropriate proof with correct conclusion is used, but multiple reasons are faulty
or multiple steps are missing or have errors.
−−
−−
BD is an altitude to AC and a reason is given, but no appropriate method is shown.
1
0
A zero response is completely incorrect, irrelevant, or incoherent or is a correct response
that was obtained by an obviously incorrect procedure.
Chapters 1–6
(pages 450–453)
Part I
Allow a total of 20 credits, 2 credits for each of the following correct answers. Allow credit if the student has
written the correct answer instead of the numeral 1, 2, 3, or 4.
6 (3) 4z
1 (3) a ab b 2
2 (1) (x 9, y 5)
7 (2)
−− −−−
−− −−
8 (2) XY ⬜ MA
3 (4) AE BE
4 (2) (x, y)
9 (1) (4, 5)
5 (4) ~r → ~p
10 (2) dilation
Part II
For each question, use the specific criteria to award a maximum of 2 credits.
11
106
Score
Explanation
2
25, and an appropriate explanation is given.
1
25, but no explanation is given.
0
A zero response is completely incorrect, irrelevant, or incoherent or is a correct response
that was obtained by an obviously incorrect procedure.
Cumulative Reviews
12
13
14
Score
Explanation
2
100, and an appropriate explanation is given.
1
100, but no explanation is given.
0
A zero response is completely incorrect, irrelevant, or incoherent or is a correct response
that was obtained by an obviously incorrect procedure.
Score
Explanation
2
a (3, 4) b (3, 4)
1
Student answers only two parts correctly.
0
A zero response is completely incorrect, irrelevant, or incoherent or is a correct response
that was obtained by an obviously incorrect procedure.
Score
c (4, 3)
d (3, 2)
Explanation
2
39 and 51, and an appropriate explanation is given.
1
39 and 51, but no explanation is given.
0
A zero response is completely incorrect, irrelevant, or incoherent or is a correct response
that was obtained by an obviously incorrect procedure.
Part III
For each question, use the specific criteria to award a maximum of 4 credits.
15
16
Score
Explanation
4
45-45-90 isosceles triangle and an appropriate explanation is given.
3
An appropriate method is used, but student does not identify the type of triangle.
2
An appropriate method is used, but multiple computational errors are made.
1
45-45-90 isosceles triangle, but no explanation is given.
0
A zero response is completely incorrect, irrelevant, or incoherent or is a correct response
that was obtained by an obviously incorrect procedure.
Score
Explanation
4
36, and an appropriate explanation is given.
3
An appropriate method is used, but one computational error is made.
2
An appropriate method is used, but multiple computational errors are made.
1
36, but no explanation is given.
0
A zero response is completely incorrect, irrelevant, or incoherent or is a correct response
that was obtained by an obviously incorrect procedure.
Chapters 1–6
107
17 Score
4
Explanation
The following proof or another appropriate proof is given.
Statements
−− −−
1. AB EB
−− −−
2. BC BD
Reasons
1. Given.
3. ⬔B ⬔B
3. Reflexive property of congruence.
4. 䉭ABC 䉭EBD
4. SAS SAS.
5. ⬔1 ⬔2
5. CPCTC.
2. Given.
3
An appropriate proof with correct conclusion is shown, but one reason is faulty and/or
one statement is missing.
2
An appropriate proof with correct conclusion is used, but multiple reasons are faulty or
multiple steps are missing or have errors.
1
⬔1 ⬔2 and a reason is given, but no appropriate method is shown.
0
A zero response is completely incorrect, irrelevant, or incoherent or is a correct response
that was obtained by an obviously incorrect procedure.
Part IV
For each question, use the specific criteria to award a maximum of 6 credits.
18
Score
Explanation
1
a K(1, 1), A(5, 2), T(3, 5)
1
b K (1, 1) A(5, 2), T (3, 5)
2
c
y
T'
T
5
4
3
A'
2
K'
1
A
K
5 4 3 2 1
1
1
2
K"
2
3
4
5
x
A"
3
4
5
108
T"
2
d r x-axis
0
A zero response is completely incorrect, irrelevant, or incoherent or is a correct response
that was obtained by an obviously incorrect procedure.
Cumulative Reviews
19
Score
6
Explanation
The following proof or another appropriate proof is given.
Statements
−− −−
1. Isosceles 䉭ABC, with AB BC
−−
2. BD bisects ⬔ABC.
1. Given.
2. Given.
3. ⬔ABD ⬔CBD
−− −−
4. BD BD
3. Definition of angle bisector.
5. 䉭ABD 䉭CBD
−−− −−−
6. AD CD
−−
−−
7. BD bisects AC.
5. SAS SAS.
4. Reflexive property of congruence.
6. CPCTC.
7. Definition of bisector.
5
An appropriate proof with correct conclusion is shown, but one reason is faulty and/or
one statement is missing.
4
A faulty conclusion or no conclusion is drawn from the statements and reasoning.
3
An appropriate proof with correct conclusion is shown, but two reasons are faulty or
two steps are missing.
2
An appropriate proof with correct conclusion is used, but more than two reasons are
faulty or more than two steps are missing or have errors.
−−
−−
BD bisects AC and a reason is given, but no appropriate method is shown.
1
0
20
Reasons
A zero response is completely incorrect, irrelevant, or incoherent or is a correct response
that was obtained by an obviously incorrect procedure.
Score
Explanation
1
a C
1
b D
2
c B
2
d C
0
A zero response is completely incorrect, irrelevant, or incoherent or is a correct
response that was obtained by an obviously incorrect procedure.
Chapters 1–7
(pages 454–456)
Part I
Allow a total of 20 credits, 2 credits for each of the following correct answers. Allow credit if the student has
written the correct answer instead of the numeral 1, 2, 3, or 4.
−− −−−
1 (3) supplementary
6 (4) AB RM
2 (4) (0, 6)
7 (2) 3(b a)
3 (1) 6
8 (1) ⬔B is the largest angle.
4 (3) I and III only
9 (1) 15
x
5 (2) 2
10 (1) _
2
Chapters 1–7
109
Part II
For each question, use the specific criteria to award a maximum of 2 credits.
11
12
13
14
Score
Explanation
2
6, and an appropriate explanation is given.
1
6, but no explanation is given.
0
A zero response is completely incorrect, irrelevant, or incoherent or is a correct response
that was obtained by an obviously incorrect procedure.
Score
Explanation
2
(7, 5), and an appropriate explanation is given.
1
(7, 5), but no explanation is given.
0
A zero response is completely incorrect, irrelevant, or incoherent or is a correct response
that was obtained by an obviously incorrect procedure.
Score
Explanation
2
144, and an appropriate explanation is given.
1
144, but no explanation is given.
0
A zero response is completely incorrect, irrelevant, or incoherent or is a correct response
that was obtained by an obviously incorrect procedure.
Score
Explanation
2
(3, 3), and an appropriate explanation is given.
1
(3, 3), but no explanation is given.
0
A zero response is completely incorrect, irrelevant, or incoherent or is a correct response
that was obtained by an obviously incorrect procedure.
Part III
For each question, use the specific criteria to award a maximum of 4 credits.
15
Score
1
Explanation
a (4, 0)
b (4, 2)
1
c (6, 3)
d (3, 6)
2
e (8, 3)
f (6, 3)
0
110
A zero response is completely incorrect, irrelevant, or incoherent or is a correct response
that was obtained by an obviously incorrect procedure.
Cumulative Reviews
16
17
Score
Explanation
4
The following proof or another appropriate proof is given.
−− −−
Right angles B and A are congruent because AB ⬜ DB and
−− −−− −−− −−−
AC ⬜ DC. BW CW. ⬔BWA ⬔CWD because vertical angles are congruent.
−−− −−−
Therefore, by ASA 䉭ABW 䉭DCW. AW DW because of CPCTC. Hence, 䉭AWD is
isosceles by definition of an isosceles triangle.
3
An appropriate proof with correct conclusion is shown, but one reason is faulty and/or
one statement is missing.
2
An appropriate proof with correct conclusion is used, but multiple reasons are faulty or
multiple steps are missing or have errors.
1
䉭AWD is isosceles and a reason is given, but no appropriate method is shown.
0
A zero response is completely incorrect, irrelevant, or incoherent or is a correct response
that was obtained by an obviously incorrect procedure.
Score
4
Explanation
The following proof or another appropriate proof is given.
Statements
Reasons
1. BC AB
1. Given.
2. m⬔BAC m⬔BCA
2. The measures of the angles opposite unequal
sides are unequal.
1 m⬔BAC _
1 m⬔BCA
3. _
2
2
−−−
4. DA bisects ⬔BAC.
3. Division postulate.
1 m⬔BAC
5. m⬔DAC _
2
−−−
6. DC bisects ⬔BCA.
5. Definition of a bisector.
1 m⬔BCA
7. m⬔DCA _
2
8. m⬔DAC m⬔DCA
7. Definition of a bisector.
9. DC DA
9. The lengths of the sides opposite two unequal
angles of a triangle are unequal.
4. Given.
6. Given.
8. Substitution postulate.
3
An appropriate proof with correct conclusion is shown, but one reason is faulty and/or
one statement is missing.
2
An appropriate proof with correct conclusion is used, but multiple reasons are faulty or
multiple steps are missing or have errors.
1
DC DA and a reason is given, but no appropriate method is shown.
0
A zero response is completely incorrect, irrelevant, or incoherent or is a correct response
that was obtained by an obviously incorrect procedure.
Chapters 1–7
111
Part IV
For each question, use the specific criteria to award a maximum of 6 credits.
18
Score
1
a (4, 1)
1
b √10
2
c y 3x 13
1 , slope of the median is 3. The slopes are negative
_
−−− d Slope MD
3
−−−
reciprocals; therefore, the median is perpendicular to MD.
2
0
19
20
Score
A zero response is completely incorrect, irrelevant, or incoherent or is a correct response
that was obtained by an obviously incorrect procedure.
Explanation
4
−−
−−−
1 x 3.
a Altitude from D to AC is x 0. Altitude from C to AD is y _
2
−−−
Altitude from A to CD is y x 3.
3
a An appropriate method is used, but one computational error is made.
2
a An appropriate method is used, but multiple computational errors are made.
1
a The altitudes are identified, but no explanation is given.
2
b (0, 3), and appropriate work is shown.
1
b (0, 3), but no work is shown.
0
A zero response is completely incorrect, irrelevant, or incoherent or is a correct response
that was obtained by an obviously incorrect procedure.
Score
6
112
Explanation
Explanation
The following proof or another appropriate proof is given.
Statements
−− −−
1. 䉭ABC, AB BC
Reasons
1. Given.
2. 䉭ABC is isosceles.
2. Definition of isosceles triangle.
3. ⬔A ⬔C
3. Definition of isosceles triangle.
4. m⬔C m⬔T
4. Definition of an exterior angle.
5. m⬔A m⬔T
5. Substitution postulate.
6. PT AP
6. Greater side opposite greater angle.
5
An appropriate proof with correct conclusion is shown, but one reason is faulty and/or
one statement is missing.
4
A faulty conclusion or no conclusion is drawn from the statements and reasoning.
3
An appropriate proof with correct conclusion is shown, but two reasons are faulty or
two steps are missing.
2
An appropriate proof with correct conclusion is used, but multiple reasons are faulty
or multiple steps are missing or have errors.
Cumulative Reviews
1
PT AP and a reason is given, but no appropriate method is shown.
0
A zero response is completely incorrect, irrelevant, or incoherent or is a correct response
that was obtained by an obviously incorrect procedure.
Chapters 1–8
(pages 457–460)
Part I
Allow a total of 20 credits, 2 credits for each of the following correct answers. Allow credit if the student has
written the correct answer instead of the numeral 1, 2, 3, or 4.
1 (3) rotation of 180
6 (1) y
2 (1) 5
7 (3) (5, 3)
3 (1) x 2y 5
8 (2) negative and less than the x-intercept of m
4 (1) 23
9 (2) 2
5 (1) {1, 2, 3}
10 (4) (9, 7)
Part II
For each question, use the specific criteria to award a maximum of 2 credits.
11
12
13
14
Score
Explanation
2
m⬔1 36, m⬔2 72, m⬔3 36, m⬔AOE 144, and an appropriate explanation is
given.
1
m⬔1 36, m⬔2 72, m⬔3 36, m⬔AOE 144, but no explanation is given.
0
A zero response is completely incorrect, irrelevant, or incoherent or is a correct response
that was obtained by an obviously incorrect procedure.
Score
Explanation
2
(0, 1), and an appropriate explanation is given.
1
(0, 1), but no explanation is given.
0
A zero response is completely incorrect, irrelevant, or incoherent or is a correct response
that was obtained by an obviously incorrect procedure.
Score
Explanation
2
(5, 1), and an appropriate explanation is given.
1
(5, 1), but no explanation is given.
0
A zero response is completely incorrect, irrelevant, or incoherent or is a correct response
that was obtained by an obviously incorrect procedure.
Score
Explanation
2
105, and an appropriate explanation is given.
1
105, but no explanation is given.
0
A zero response is completely incorrect, irrelevant, or incoherent or is a correct response
that was obtained by an obviously incorrect procedure.
Chapters 1–8
113
Part III
For each question, use the specific criteria to award a maximum of 4 credits.
15
16
17
Score
4
2 : 3, and an appropriate explanation is given.
3
An appropriate method is used, but a single arithmetical error is made.
2
An appropriate method is used, but multiple arithmetical errors are made.
1
2 : 3, but no explanation is given.
0
A zero response is completely incorrect, irrelevant, or incoherent or is a correct response
that was obtained by an obviously incorrect procedure.
Score
Explanation
4
x 7, m⬔P 62, m⬔Q 28, and an appropriate explanation is given.
3
An appropriate method is used, but a single arithmetical error is made.
2
An appropriate method is used, but multiple arithmetical errors are made.
1
x 7, m⬔P 62, m⬔Q 28, but no explanation is given.
0
A zero response is completely incorrect, irrelevant, or incoherent or is a correct response
that was obtained by an obviously incorrect procedure.
Score
4
114
Explanation
Explanation
The following proof or another appropriate proof is given.
Statements
−−− −− −− −− −− −−
1. AD BC, DE CE, EA EB
Reasons
1. Given.
2. 䉭ADE 䉭BCE
2. SSS SSS.
3. ⬔1 ⬔2
3. CPCTC.
4. ⬔3 ⬔4
5. ⬔1 ⬔3 ⬔2 ⬔4
4. Base angles of an isosceles triangle are
congruent.
5. Addition postulate.
6. ⬔DAB ⬔CBA
6. Partition postulate.
3
An appropriate proof with correct conclusion is shown, but one reason is faulty and/or
one statement is missing.
2
An appropriate proof with correct conclusion is used, but multiple reasons are faulty or
multiple steps are missing or have errors.
1
⬔DAB ⬔CBA and a reason is given, but no appropriate method is shown.
0
A zero response is completely incorrect, irrelevant, or incoherent or is a correct response
that was obtained by an obviously incorrect procedure.
Cumulative Reviews
Part IV
For each question, use the specific criteria to award a maximum of 6 credits.
18
Score
6
Explanation
2. If two sides are congruent, the triangle is isosceles.
3. Base angles of isosceles triangles are congruent.
4. Exterior angle of a triangle is greater than either nonadjacent interior angle.
5. Substitution postulate.
6. Longest side of a triangle is opposite the angle with the largest measure.
5
One reason is faulty or missing.
4
Two reasons are faulty or missing.
3
Three reasons are faulty or missing.
2
More than three reasons are faulty or missing.
0
A zero response is completely incorrect, irrelevant, or incoherent or is a correct response
that was obtained by an obviously incorrect procedure.
Check students’ graphs. The coordinates are listed below.
19
20
Score
Explanation
1
a A(6, 6), B(12, 6), C(6, 15)
2
b A(6, 6), B(12, 6), C(6, 15)
2
c A
(6, 4), B
(6, 10), C
(15, 4)
1
d D3
0
A zero response is completely incorrect, irrelevant, or incoherent or is a correct response
that was obtained by an obviously incorrect procedure.
Score
Explanation
1
a Student graphs points A and B correctly.
3
b x 4 and an appropriate explanation is given. Student graphs point C correctly.
2
c 0, and an appropriate explanation is given.
1
Correct answers for b and c, but no explanation is given.
0
A zero response is completely incorrect, irrelevant, or incoherent or is a correct response
that was obtained by an obviously incorrect procedure.
Chapters 1–8
115
Chapters 1–9
(pages 461–463)
Part I
Allow a total of 20 credits, 2 credits for each of the following correct answers. Allow credit if the student has
written the correct answer instead of the numeral 1, 2, 3, or 4.
1 (3) right
6 (1) 34
−−
2 (4) AE bisects ⬔BAC.
7 (3) SSS
3 (4) 4, 5, 6
8 (1) 290
4 (1) (5, 3)
9 (3) 3,600
5 (2) congruent
10 (1) 30
Part II
For each question, use the specific criteria to award a maximum of 2 credits.
11
12
13
14
116
Score
Explanation
2
x is divisible by 2, and an appropriate explanation is given.
1
x is divisible by 2, but no explanation is given.
0
A zero response is completely incorrect, irrelevant, or incoherent or is a correct response
that was obtained by an obviously incorrect procedure.
Score
Explanation
2
155, and an appropriate explanation is given.
1
155, but no explanation is given.
0
A zero response is completely incorrect, irrelevant, or incoherent or is a correct response
that was obtained by an obviously incorrect procedure.
Score
Explanation
2
(9, 2), and an appropriate explanation is given.
1
(9, 2), but no explanation is given.
0
A zero response is completely incorrect, irrelevant, or incoherent or is a correct response
that was obtained by an obviously incorrect procedure.
Score
Explanation
2
(4, 2), and an appropriate explanation is given.
1
(4, 2), but no explanation is given.
0
A zero response is completely incorrect, irrelevant, or incoherent or is a correct response
that was obtained by an obviously incorrect procedure.
Cumulative Reviews
Part III
For each question, use the specific criteria to award a maximum of 4 credits.
15
Score
4
16
17
Explanation
a True
b True
c False
d True
e False
f True
3
Student answers all but one part correctly.
2
Student answers two or three parts incorrectly.
1
Student answers four or five parts incorrectly.
0
A zero response is completely incorrect, irrelevant, or incoherent or is a correct response
that was obtained by an obviously incorrect procedure.
Score
Explanation
4
18, and an appropriate explanation is given.
3
An appropriate method is used, but a single arithmetical error is made.
2
An appropriate method is used, but multiple arithmetical errors are made.
1
18, but no explanation is given.
0
A zero response is completely incorrect, irrelevant, or incoherent or is a correct response
that was obtained by an obviously incorrect procedure.
Score
4
Explanation
a Yes
b Yes
c Yes
d No
e No
f Yes
3
Student answers all but one part correctly.
2
Student answers two or three parts incorrectly.
1
Student answers four or five parts incorrectly.
0
A zero response is completely incorrect, irrelevant, or incoherent or is a correct response
that was obtained by an obviously incorrect procedure.
Part IV
For each question, use the specific criteria to award a maximum of 6 credits.
18
Score
Explanation
6
, AC √
AB √
85 , BC √85
68 , and an appropriate explanation is given.
5
An appropriate method is shown, but one computational mistake is made.
4
An appropriate method is shown, but two computational mistakes are made.
3
An appropriate method is shown, but wrong sides of the triangle are shown congruent.
2
An appropriate method is shown, but more than two computational mistakes are
made.
1
An appropriate method is shown, but no conclusion is given.
0
A zero response is completely incorrect, irrelevant, or incoherent or is a correct response
that was obtained by an obviously incorrect procedure.
Chapters 1–9
117
19
Score
6
Explanation
The following proof or another appropriate proof is given.
Statements
1. 䉭ABC is equilateral.
1. Given.
2. m⬔BAC 60
2. Interior angles of an equilateral
triangle are 60.
3. ⬔BCA and ⬔BCE are a linear pair.
3. Definition of linear pair.
4. ⬔BCE is supplementary to ⬔BCA.
4. If two angles form a linear pair, then
they are supplementary.
5. m⬔BCA m⬔BCE 180
5. Definition of supplementary angles.
6. m⬔BCE 120
−−−
7. CD bisects ⬔BCE.
1 m⬔BCE
8. m⬔DCE _
2
9. m⬔DCE 60
6. Subtraction postulate.
7. Given.
8. Definition of angle bisector.
9. Substitution.
10. m⬔DCE m⬔BAC
10. Substitution.
11. ⬔DCE ⬔BAC
11. Angles with equal measures are
congruent.
−−
AB
12. CD
12. Lines with congruent corresponding
angles are parallel.
5
An appropriate proof with correct conclusion is shown, but one reason is faulty and/or
one statement is missing.
4
A faulty conclusion or no conclusion is drawn from the statements and reasoning.
3
An appropriate proof with correct conclusion is shown, but two reasons are faulty or
two steps are missing.
2
An appropriate proof with correct conclusion is used, but more than two reasons are
faulty or more than two steps are missing or have errors.
−−
AB and a reason is given, but no appropriate method is shown.
CD
1
0
118
Reasons
A zero response is completely incorrect, irrelevant, or incoherent or is a correct response
that was obtained by an obviously incorrect procedure.
Cumulative Reviews
20
Score
6
Explanation
The following proof or another appropriate proof is given.
Statements
−− −−−
1. BC AD
−− −−−
2. BC AD
Reasons
1. Given.
2. Given.
3. ⬔EAD ⬔FCB
−− −−
4. BF ED
4. Given.
5. ⬔BFC ⬔DEA
5. Alternate interior angles are congruent.
6. 䉭BFC 䉭DEA
−− −−
7. BF ED
6. AAS AAS.
3. Alternate interior angles are congruent.
7. CPCTC.
5
An appropriate proof with correct conclusion is shown, but one reason is faulty and/or
one statement is missing.
4
A faulty conclusion or no conclusion is drawn from the statements and reasoning.
3
An appropriate proof with correct conclusion is shown, but two reasons are faulty or
two steps are missing.
2
An appropriate proof with correct conclusion is used, but more than two reasons are
faulty or more than two steps are missing or have errors.
−− −−
BF ED and a reason is given, but no appropriate method is shown.
1
0
A zero response is completely incorrect, irrelevant, or incoherent or is a correct response
that was obtained by an obviously incorrect procedure.
Chapters 1–10
(pages 464–466)
Part I
Allow a total of 20 credits, 2 credits for each of the following correct answers. Allow credit if the student has
written the correct answer instead of the numeral 1, 2, 3, or 4.
10
1x _
1 (2) 360
7 (1) y _
3
3
2 (4) 140
8 (4) The diagonals bisect the angles of
3 (1) 22
the parallelogram.
4 (3) y x
9 (1) adjacent
5 (3) 8
10 (3) m⬔4 m⬔B
6 (3) a rhombus
Part II
For each question, use the specific criteria to award a maximum of 2 credits.
11
Score
Explanation
2
120, and an appropriate explanation is given.
1
120, but no explanation is given.
0
A zero response is completely incorrect, irrelevant, or incoherent or is a correct response
that was obtained by an obviously incorrect procedure.
Chapters 1–10
119
12
13
14
Score
Explanation
2
31, and an appropriate explanation is given.
1
31, but no explanation is given.
0
A zero response is completely incorrect, irrelevant, or incoherent or is a correct response
that was obtained by an obviously incorrect procedure.
Score
Explanation
2
7 √
2 , and an appropriate explanation is given.
1
7 √2, but no explanation is given.
0
A zero response is completely incorrect, irrelevant, or incoherent or is a correct response
that was obtained by an obviously incorrect procedure.
Score
Explanation
2
9, and an appropriate explanation is given.
1
9, but no explanation is given.
0
A zero response is completely incorrect, irrelevant, or incoherent or is a correct response
that was obtained by an obviously incorrect procedure.
Part III
For each question, use the specific criteria to award a maximum of 4 credits.
15
16
120
Score
Explanation
4
6, and an appropriate explanation is given.
3
An appropriate method is used, but a single computational error is made.
2
An appropriate method is used, but multiple computational errors are made.
1
6, but no explanation is given.
0
A zero response is completely incorrect, irrelevant, or incoherent or is a correct response
that was obtained by an obviously incorrect procedure.
Score
Explanation
4
A(3, 6), B(2, 1), C(9, 1), and the transformations are performed correctly.
3
Student arrives at the correct answer, but performs one incorrect transformation.
2
Student performs transformations in the wrong order.
1
A(3, 6), B(2, 1), C(9, 1), but no appropriate explanation is given.
0
A zero response is completely incorrect, irrelevant, or incoherent or is a correct response
that was obtained by an obviously incorrect procedure.
Cumulative Reviews
17
Score
Explanation
4
51, and an appropriate explanation is given.
3
An appropriate method is used, but a single computational error is made.
2
An appropriate method is used, but multiple computational errors are made.
1
51, but no explanation is given.
0
A zero response is completely incorrect, irrelevant, or incoherent or is a correct response
that was obtained by an obviously incorrect procedure.
Part IV
For each question, use the specific criteria to award a maximum of 6 credits.
18
Score
3
19
Explanation
−−
1 , slope −− _
2 , slope −− _
1 , TD
_
−−− is a vertical line. Since
a Slope DA
RT
AR
3
3
3
−−− −−
two sides have the same slope, DA RT. DART is a trapezoid.
0
A zero response is completely incorrect, irrelevant, or incoherent or is a correct
response that was obtained by an obviously incorrect procedure.
3
b AK TA, and an appropriate explanation is given.
0
A zero response is completely incorrect, irrelevant, or incoherent or is a correct response
that was obtained by an obviously incorrect procedure.
Score
3
Explanation
a The following proof or another appropriate proof is given.
Statements
−−− −−
1. AD BE
−− −−
2. AE BD
2. Given.
3. DE DE
3. Reflexive property of congruence.
4. 䉭ADE 䉭BED
4. SSS SSS.
5. ⬔BDE ⬔AED
5. CPCTC.
___
___
Reasons
1. Given.
1
⬔BDE ⬔AED and a reason is given, but no appropriate method is shown.
0
A zero response is completely incorrect, irrelevant, or incoherent or is a correct response
that was obtained by an obviously incorrect procedure.
3
b The following proof or another appropriate proof is given.
Statements
−−− −−
1. AD BE
−− −−
2. AE BD
−− −−
3. AB AB
Reasons
1. Given.
2. Given.
3. Reflexive property of congruence.
4. 䉭ADB 䉭BEA
4. SSS SSS.
5. ⬔FAB ⬔FBA
5. CPCTC.
1
⬔FAB ⬔FBA and a reason is given, but no appropriate method is shown.
0
A zero response is completely incorrect, irrelevant, or incoherent or is a correct response
that was obtained by an obviously incorrect procedure.
Chapters 1–10
121
20
Score
6
Explanation
The following proof or another appropriate proof is given.
Statements
Reasons
1. Quadrilateral ABCD and
−−−−
diagonal AFEC
−− −− −− −−
2. BF ⬜ AC, DE ⬜ AC
1. Given.
3. ⬔BFA and ⬔DEC are right
angles.
3. Definition of perpendicular lines.
4. ⬔BFA ⬔DEC
−− −− −− −−
5. BF DE, AE CF
4. Right angles are congruent.
6. 䉭ADE 䉭CBF
−−− −−
7. AD BC
8. ⬔DAE ⬔BCF
−−− −−
9. AD BC
10. ABCD is a parallelogram.
2. Given.
5. Given.
6. SAS SAS.
7. CPCTC.
8. CPCTC.
9. Alternate interior angles of two parallel lines
are congruent.
10. Definition of a parallelogram.
5
An appropriate proof with correct conclusion is shown, but one reason is faulty and/or
one statement is missing.
4
A faulty conclusion or no conclusion is drawn from the statements and reasoning.
3
An appropriate proof with correct conclusion is shown, but two reasons are faulty or
two steps are missing.
2
An appropriate proof with correct conclusion is used, but more than two reasons are
faulty or more than two steps are missing or have errors.
1
ABCD is a parallelogram and a reason is given, but no appropriate method is shown.
0
A zero response is completely incorrect, irrelevant, or incoherent or is a correct response
that was obtained by an obviously incorrect procedure.
Chapters 1–11
(pages 467–469)
Part I
Allow a total of 20 credits, 2 credits for each of the following correct answers. Allow credit if the student has
written the correct answer instead of the numeral 1, 2, 3, or 4.
2
b
6 (3) 160 cm
1 (1) _
a
7 (1) 2 √6
2 (3) equilateral and equiangular
3
3 (3) 120
8 (3) _
−−
5
4 (1) AC
9 (3) The sphere is inscribed in the cube.
5 (2) 9
10 (1) parallelogram
122
Cumulative Reviews
Part II
For each question, use the specific criteria to award a maximum of 2 credits.
11
12
13
14
Score
Explanation
2
x 116, y 32, and an appropriate explanation is given.
1
x 116, y 32, but no explanation is given.
0
A zero response is completely incorrect, irrelevant, or incoherent or is a correct response
that was obtained by an obviously incorrect procedure.
Score
Explanation
2
n 1, and an appropriate explanation is given.
1
n 1, but no explanation is given.
0
A zero response is completely incorrect, irrelevant, or incoherent or is a correct response
that was obtained by an obviously incorrect procedure.
Score
Explanation
2
116.5, and an appropriate explanation is given.
1
116.5, but no explanation is given.
0
A zero response is completely incorrect, irrelevant, or incoherent or is a correct response
that was obtained by an obviously incorrect procedure.
Score
Explanation
2
50, and an appropriate explanation is given.
1
50, but no explanation is given.
0
A zero response is completely incorrect, irrelevant, or incoherent or is a correct response
that was obtained by an obviously incorrect procedure.
Part III
For each question, use the specific criteria to award a maximum of 4 credits.
15
Score
Explanation
4
10 , and an appropriate explanation is given.
3 √
3
An appropriate method is used, but a single arithmetical error is made.
2
An appropriate method is used, but multiple arithmetical errors are made.
1
3 √
10 , but no explanation is given.
0
A zero response is completely incorrect, irrelevant, or incoherent or is a correct response
that was obtained by an obviously incorrect procedure.
Chapters 1–11
123
16
17
124
Score
Explanation
4
DN √
106 , NA √
106 , making 䉭DNA isosceles.
9
5
_
_
−−− −−− and slope NA
. Since the slopes are negative reciprocals of each
Slope DN
5
9
−−− −−−
other, DN ⬜ NA. Therefore, 䉭DNA is an isosceles right triangle.
3
An appropriate method is used, but a single computational error is made.
2
An appropriate method is used, but 䉭DNA is not proven to be an isosceles right
triangle.
1
An appropriate method is used, but multiple computational errors are made.
0
A zero response is completely incorrect, irrelevant, or incoherent or is a correct response
that was obtained by an obviously incorrect procedure.
Score
Explanation
4
6 √
, and an appropriate explanation is given.
3
An appropriate method is used, but a single computational error is made.
2
An appropriate method is used, but multiple computational errors are made.
1
6 √
, but no explanation is given.
0
A zero response is completely incorrect, irrelevant, or incoherent or is a correct response
that was obtained by an obviously incorrect procedure.
Cumulative Reviews
Part IV
For each question, use the specific criteria to award a maximum of 6 credits.
18 Score
6
Explanation
The following proof or another appropriate proof is given.
Statements
−−− −− −−− −−
1. DA ⬜ AB, DC ⬜ CB
5
19
Reasons
1. Given.
2. ⬔BAD and ⬔BCD are right angles.
2. Definition of perpendicular lines.
3. ⬔BAD ⬔BCD
3. Right angles are congruent.
4. ⬔1 ⬔2
4. Given.
5. ⬔BAE ⬔BCE
−− −−
6. BD ⬜ AC at E
5. Subtraction postulate.
7. ⬔BEA and ⬔BEC are right angles.
7. Definition of perpendicular lines.
8. ⬔BEA ⬔BEC
−− −−
9. BE BE
8. Right angles are congruent.
6. Given.
9. Reflexive property of congruence.
10. 䉭ABE 䉭CBE
10. AAS AAS.
11. CPCTC.
11. ⬔3 ⬔4
An appropriate proof with correct conclusion is shown, but one reason is faulty and/
or one statement is missing.
4
A faulty conclusion or no conclusion is drawn from the statements and reasoning.
3
An appropriate proof with correct conclusion is shown, but two reasons are faulty or
two steps are missing.
2
An appropriate proof with correct conclusion is used, but more than two reasons are
faulty or more than two steps are missing or have errors.
1
⬔3 ⬔4 and a reason is given, but no appropriate method is shown.
0
A zero response is completely incorrect, irrelevant, or incoherent or is a correct
response that was obtained by an obviously incorrect procedure.
Score
Explanation
2
a Student proves that MIKE is a rhombus by showing that
40 or by some other appropriate method.
MI IK KE EM √
1
a The correct conclusion is reached and a reason is given, but no appropriate method
is shown.
2
b Student shows that MIKE is not a square by showing that the
1 and the slope −− 3.
_
−− diagonals are not perpendicular. Slope MI
IK
3
1
b The correct conclusion is reached and a reason is given, but no appropriate method
is shown.
2
c 32, and an appropriate explanation is given.
1
c 32, but no appropriate explanation is given.
0
A zero response is completely incorrect, irrelevant, or incoherent or is a correct response
that was obtained by an obviously incorrect procedure.
Chapters 1–11
125
20
Score
Explanation
2
a 3, and an appropriate explanation is given.
1
3, but no explanation is given.
0
A zero response is completely incorrect, irrelevant, or incoherent or is a correct
response that was obtained by an obviously incorrect procedure.
2
b 36, and an appropriate explanation is given.
1
36, but no explanation is given.
0
A zero response is completely incorrect, irrelevant, or incoherent or is a correct
response that was obtained by an obviously incorrect procedure.
1 , and an appropriate explanation is given.
c _
4
1 , but no explanation is given.
_
4
A zero response is completely incorrect, irrelevant, or incoherent or is a correct response
that was obtained by an obviously incorrect procedure.
2
1
0
Chapters 1–12
(pages 470–472)
Part I
Allow a total of 20 credits, 2 credits for each of the following correct answers. Allow credit if the student has
written the correct answer instead of the numeral 1, 2, 3, or 4.
1 (3) It has a slope of 0.
2 (2) are congruent
3 (3) 50
4 (3) cylinder
5 (4) 130
2
6 (3) 16x
2
7 (2) 10 √
8 (1) 1 : 3
4
9 (1) _
25
10 (3) 4x y 2
Part II
For each question, use the specific criteria to award a maximum of 2 credits.
11
126
Score
Explanation
2
225, and an appropriate explanation is given.
1
225, but no explanation is given.
0
A zero response is completely incorrect, irrelevant, or incoherent or is a correct response
that was obtained by an obviously incorrect procedure.
Cumulative Reviews
12
13
Score
2
(0, 4) and (6, 0), and an appropriate explanation is given.
1
(0, 4) and (6, 0), but no explanation is given.
0
A zero response is completely incorrect, irrelevant, or incoherent or is a correct response
that was obtained by an obviously incorrect procedure.
Score
1
Explanation
9
a _, and an appropriate explanation is given.
16
0
A zero response is completely incorrect, irrelevant, or incoherent or is a correct response
that was obtained by an obviously incorrect procedure.
1
27 , and an appropriate explanation is given.
b _
64
A zero response is completely incorrect, irrelevant, or incoherent or is a correct response
that was obtained by an obviously incorrect procedure.
0
14
Explanation
Score
Explanation
2
102, and an appropriate explanation is given.
1
102, but no explanation is given.
0
A zero response is completely incorrect, irrelevant, or incoherent or is a correct response
that was obtained by an obviously incorrect procedure.
Part III
For each question, use the specific criteria to award a maximum of 4 credits.
15
16
Score
Explanation
2
a 22, and an appropriate explanation is given.
1
a 22, but no appropriate explanation is given.
0
A zero response is completely incorrect, irrelevant, or incoherent or is a correct response
that was obtained by an obviously incorrect procedure.
2
b 78, and an appropriate explanation is given.
1
b 78, but no explanation is given.
0
A zero response is completely incorrect, irrelevant, or incoherent or is a correct response
that was obtained by an obviously incorrect procedure.
Score
Explanation
4
2 √
3 , and an appropriate explanation is given.
3
An appropriate method is used, but a single computational error is made.
2
An appropriate method is used, but multiple computational errors are made.
1
2 √
3 , but no explanation is given.
0
A zero response is completely incorrect, irrelevant, or incoherent or is a correct response
that was obtained by an obviously incorrect procedure.
Chapters 1–12
127
17
Score
Explanation
4
6 √
5 , and an appropriate explanation is given.
3
An appropriate method is used, but a single computational error is made.
2
An appropriate method is used, but multiple computational errors are made.
1
6 √
5 , but no explanation is given.
0
A zero response is completely incorrect, irrelevant, or incoherent or is a correct response
that was obtained by an obviously incorrect procedure.
Part IV
For each question, use the specific criteria to award a maximum of 6 credits.
18
128
Score
Explanation
6
Student uses an appropriate method, such as showing that both pairs of opposite sides
b , slope −− _
b.
_
−−− 0, slope −−− 0; slope −−− are parallel, by showing that slope AD
ME
ED
MA
a
a
5
An appropriate method is used, but one computational error is made.
4
An appropriate method is used, but two computational errors are made.
3
An appropriate method is used, but three computational errors are made.
2
More than three computational errors are made.
1
An appropriate method is used, but does not prove that MADE is a parallelogram.
0
A zero response is completely incorrect, irrelevant, or incoherent or is a correct response
that was obtained by an obviously incorrect procedure.
Cumulative Reviews
19
Score
6
Explanation
The following proof or another appropriate proof is given.
Statements
1. 䉭ABC is isosceles.
−− −−
2. AB AC
Reasons
1. Given.
2. Definition of isosceles triangle.
3. 䉭AMC is isosceles.
−−− −−−
4. AM CM
−−− −−−
5. PM QM
−−− −−− −−− −−−
6. AM QM CM PM
−−− −−
7. AQ CP
5. Given.
8. ⬔BAC ⬔BCA
8. Isosceles triangle theorem.
9. ⬔MAC ⬔MCA
9. Isosceles triangle theorem.
3. Given.
4. Definition of isosceles triangle.
6. Addition postulate.
7. Partition postulate.
10. ⬔BAC ⬔MAC ⬔BCA ⬔MCA
10. Subtraction postulate.
11. ⬔BAM ⬔BCP
11. Partition postulate.
12. 䉭ABQ 䉭CBP
12. SAS SAS.
5
An appropriate proof with correct conclusion is shown, but one reason is faulty and/or
one statement is missing.
4
A faulty conclusion or no conclusion is drawn from the statements and reasoning.
3
An appropriate proof with correct conclusion is shown, but two reasons are faulty or
two steps are missing.
2
An appropriate proof with correct conclusion is used, but more than two reasons are
faulty or more than two steps are missing or have errors.
1
䉭ABQ 䉭CBP and a reason is given, but no appropriate method is shown.
0
A zero response is completely incorrect, irrelevant, or incoherent or is a correct response
that was obtained by an obviously incorrect procedure.
Chapters 1–12
129
20
Score
6
Explanation
The following proof or another appropriate proof is given.
Statements
Reasons
1. Parallelogram ABCD
1. Given.
2. ⬔A ⬔C
2. Opposite angles of a parallelogram are
congruent.
3. m⬔1 m⬔2
3. Given.
4. ⬔1 ⬔2
4. Angles with equal measures are
congruent.
5. ⬔1 and ⬔4 are a linear pair.
⬔2 and ⬔3 are a linear pair.
5. Definition of linear pair.
6. ⬔1 and ⬔4 are supplementary. ⬔2 and
⬔3 are supplementary.
6. Two angles that form a linear pair are
supplementary.
7. ⬔3 ⬔4
7. Supplements of congruent angles are
congruent.
8. 䉭AFE 䉭CHG
8. AA.
GH
FE _
9. _
FA
CH
9. Corresponding parts of similar
triangles are in proportion.
10. FE CH GH FA
10. The product of means equals the
product of extremes.
5
An appropriate proof with correct conclusion is shown, but one reason is faulty and/or
one statement is missing.
4
A faulty conclusion or no conclusion is drawn from the statements and reasoning.
3
An appropriate proof with correct conclusion is shown, but two reasons are faulty or
two steps are missing.
2
An appropriate proof with correct conclusion is used, but more than two reasons are
faulty or more than two steps are missing or have errors.
1
FE CH GH FA and a reason is given, but no appropriate method is shown.
0
A zero response is completely incorrect, irrelevant, or incoherent or is a correct response
that was obtained by an obviously incorrect procedure.
Chapters 1–13
(pages 473–475)
Part I
Allow a total of 20 credits, 2 credits for each of the following correct answers. Allow credit if the student has
written the correct answer instead of the numeral 1, 2, 3, or 4.
1 (4) 䉭ABD 䉭CDB
6 (3) R 270
2 (2) a median
7 (1) 30
3 (1) 12
8 (1) 12
4 (1) 1 : 2
9 (3) III and IV
−− −−
5 (1) 2
10 (4) TA TO
130
Cumulative Reviews
Part II
For each question, use the specific criteria to award a maximum of 2 credits.
11
12
13
14
Score
Explanation
2
(16, 14), and an appropriate explanation is given.
1
(16, 14), but no explanation is given.
0
A zero response is completely incorrect, irrelevant, or incoherent or is a correct response
that was obtained by an obviously incorrect procedure.
Score
Explanation
2
55, and an appropriate explanation is given.
1
55, but no explanation is given.
0
A zero response is completely incorrect, irrelevant, or incoherent or is a correct response
that was obtained by an obviously incorrect procedure.
Score
Explanation
2
38, and an appropriate explanation is given.
1
38, but no explanation is given.
0
A zero response is completely incorrect, irrelevant, or incoherent or is a correct response
that was obtained by an obviously incorrect procedure.
Score
Explanation
2
30.5, and an appropriate explanation is given.
1
30.5, but no explanation is given.
0
A zero response is completely incorrect, irrelevant, or incoherent or is a correct response
that was obtained by an obviously incorrect procedure.
Part III
For each question, use the specific criteria to award a maximum of 4 credits.
15
Score
Explanation
4
y x 3, and an appropriate explanation is given.
3
An appropriate method is used, but a single computational error is made.
2
An appropriate method is used, but the equation of a parallel line is given.
1
y x 3, but no explanation is given.
0
A zero response is completely incorrect, irrelevant, or incoherent or is a correct response
that was obtained by an obviously incorrect procedure.
Chapters 1–13
131
16
17
Score
Explanation
4
m⬔ADE 93, m⬔BGA 87, m⬔ABC 87, and an appropriate explanation is given.
3
An appropriate method is used, but a single computational error is made.
2
An appropriate method is used, but one of the angle measures found is incorrect.
1
m⬔ADE 93, m⬔BGA 87, m⬔ABC 87, but no explanation is given.
0
A zero response is completely incorrect, irrelevant, or incoherent or is a correct response
that was obtained by an obviously incorrect procedure.
Score
4
3
2
1
0
Explanation
r
h_
, and an appropriate explanation is given.
An appropriate method is used, but a single computational error is made.
An appropriate method is used, but h is found in terms of r and s.
r
h_
, but no explanation is given.
A zero response is completely incorrect, irrelevant, or incoherent or is a correct response
that was obtained by an obviously incorrect procedure.
Part IV
For each question, use the specific criteria to award a maximum of 6 credits.
18
132
Score
Explanation
6
h 12, and an appropriate explanation is given.
5
An appropriate method is used, but one computational error is made.
4
An appropriate method is used, but two computational errors are made.
3
An appropriate method is used, but three computational errors are made.
2
More than three computational errors are made.
1
h 12, but no explanation is given.
0
A zero response is completely incorrect, irrelevant, or incoherent or is a correct response
that was obtained by an obviously incorrect procedure.
Cumulative Reviews
19
Score
6
Explanation
The following proof or another appropriate proof is given.
Statements
Reasons
1. Parallelogram ABCD;
−−− −−−
EAB; DCF
−− −−−
2. AB DC
−− −−
3. EB DF
1. Given.
4. ⬔AEH ⬔CFG
4. Alternate interior angles are congruent.
5. ⬔BAH ⬔BCD
5. Opposite angles of a parallelogram are
congruent.
6. ⬔GCF and ⬔BCD are a linear
pair. ⬔HAE and ⬔BAH are a
linear pair.
6. Definition of linear pair.
7. ⬔GCF and ⬔BCD are supplementary. ⬔HAE and ⬔BAH are
supplementary.
7. Two angles that form a linear pair are
supplementary.
8. ⬔GCF ⬔HAE
8. Supplements of congruent angles are
congruent.
9. a 䉭EAH 䉭FCG
FC
EA _
10. _
AH
CG
11. b EA CG AH CG
2. Definition of a parallelogram.
3. Collinearity.
9. AA.
10. Corresponding parts of similar triangles are in
proportion.
11. The product of means equals the product of
extremes.
5
An appropriate proof with correct conclusion is shown, but one reason is faulty and/or
one statement is missing.
4
An appropriate proof with correct conclusion is shown for part a only.
4
A faulty conclusion or no conclusion is drawn from the statements and reasoning.
3
An appropriate proof with correct conclusion is shown, but two reasons are faulty or
two steps are missing.
2
An appropriate proof with correct conclusion is used, but more than two reasons are
faulty or more than two steps are missing or have errors.
1
䉭EAH 䉭FCG and EA CG AH CG, and a reason is given, but no appropriate
method is shown.
0
A zero response is completely incorrect, irrelevant, or incoherent or is a correct response
that was obtained by an obviously incorrect procedure.
Chapters 1–13
133
20
Score
Explanation
1
72, and an appropriate explanation is given.
a mBC
0
A zero response is completely incorrect, irrelevant, or incoherent or is a correct response
that was obtained by an obviously incorrect procedure.
1
b m⬔EDC 108, and an appropriate explanation is given.
0
A zero response is completely incorrect, irrelevant, or incoherent or is a correct response
that was obtained by an obviously incorrect procedure.
1
c m⬔BCR 36, and an appropriate explanation is given.
0
A zero response is completely incorrect, irrelevant, or incoherent or is a correct response
that was obtained by an obviously incorrect procedure.
1
d m⬔AKB 72, and an appropriate explanation is given.
0
A zero response is completely incorrect, irrelevant, or incoherent or is a correct response
that was obtained by an obviously incorrect procedure.
2
e m⬔RF 72, and an appropriate explanation is given.
0
A zero response is completely incorrect, irrelevant, or incoherent or is a correct response
that was obtained by an obviously incorrect procedure.
Chapters 1–14
(pages 476–479)
Part I
Allow a total of 20 credits, 2 credits for each of the following correct answers. Allow credit if the student has
written the correct answer instead of the numeral 1, 2, 3, or 4.
1
2
3
4
5
(1) x 3
(3) 3 : 5
(3) 125
(2) 45
2
(3) 2 √
6
7
8
9
10
(2) 8.8
(2) 2x 3
(1) {C, D, G, H}
(4) y 3x 5
(4) similar triangles
Part II
For each question, use the specific criteria to award a maximum of 2 credits.
11
12
134
Score
Explanation
2
720, and an appropriate explanation is given.
1
720, but no explanation is given.
0
A zero response is completely incorrect, irrelevant, or incoherent or is a correct response
that was obtained by an obviously incorrect procedure.
Score
Explanation
2
144, and an appropriate explanation is given.
1
144, but no explanation is given.
0
A zero response is completely incorrect, irrelevant, or incoherent or is a correct response
that was obtained by an obviously incorrect procedure.
Cumulative Reviews
13
14
Score
Explanation
2
13, and an appropriate explanation is given.
1
13, but no explanation is given.
0
A zero response is completely incorrect, irrelevant, or incoherent or is a correct response
that was obtained by an obviously incorrect procedure.
Score
Explanation
2
80, and an appropriate explanation is given.
1
80, but no explanation is given.
0
A zero response is completely incorrect, irrelevant, or incoherent or is a correct response
that was obtained by an obviously incorrect procedure.
Part III
For each question, use the specific criteria to award a maximum of 4 credits.
15
16
Score
4
Each side is 2b, and an appropriate explanation is given.
3
An appropriate method is used, but a single computational error is made.
2
An appropriate method is used, but h is found in terms of r and s.
1
Each side is 2b, but no explanation is given.
0
A zero response is completely incorrect, irrelevant, or incoherent or is a correct response
that was obtained by an obviously incorrect procedure.
Score
4
3
2
1
0
17
Explanation
Score
Explanation
3
_
, and an appropriate explanation is given.
10
An appropriate method is used, but a single computational error is made.
An appropriate method is used, but h is found in terms of r and s.
3
_
, but no explanation is given.
10
A zero response is completely incorrect, irrelevant, or incoherent or is a correct response
that was obtained by an obviously incorrect procedure.
Explanation
4
30,776.3 in. 3, and an appropriate explanation is given.
3
An appropriate method is used, but a single computational error is made.
2
An appropriate method is used, but h is found in terms of r and s.
1
30,776.3 in. 3, but no explanation is given.
0
A zero response is completely incorrect, irrelevant, or incoherent or is a correct response
that was obtained by an obviously incorrect procedure.
Chapters 1–14
135
Part IV
For each question, use the specific criteria to award a maximum of 6 credits.
a
y
6
x
10
y
9
2x
3
Explanation
8
7
Score
7
3y
18
6
5
4
3
2
1
7 6 5 4 3 2 1
1
y 2
1
2
3
4
5
6
7
8
9 10
x
2
3
4
5
6
7
19
136
0
A zero response is completely incorrect, irrelevant, or incoherent or is a correct response
that was obtained by an obviously incorrect procedure.
3
b 45, and an appropriate explanation is given.
2
An appropriate method is used, but a single computational error is made.
1
45, but no explanation is given.
0
A zero response is completely incorrect, irrelevant, or incoherent or is a correct response
that was obtained by an obviously incorrect procedure.
Score
Explanation
4
−−− 1 and slope −−− 1. Since the slopes are negative reciprocals of each
a Slope AX
ME
−−−
other, the lines are perpendicular. The midpoint of AX is
−−−
(4, 3), which is point E. Therefore, ME is the perpendicular bisector.
3
An appropriate method is used, but a single arithmetical error is made.
2
An appropriate method is used to show the lines are perpendicular, but no midpoint is
found.
0
A zero response is completely incorrect, irrelevant, or incoherent or is a correct response
that was obtained by an obviously incorrect procedure.
2
b Shows that 䉭MAX is isosceles by MX MA √
68 , and an appropriate explanation
is given.
1
An appropriate method is used, but a single computational error is made.
0
A zero response is completely incorrect, irrelevant, or incoherent or is a correct response
that was obtained by an obviously incorrect procedure.
Cumulative Reviews
20
Score
6
Explanation
The following proof or another appropriate proof is given.
Statements
Reasons
CE
1. AE
1. Given.
2. AE CE
2. Congruent arcs have congruent chords.
3. ⬔ABE ⬔EBD
3. Congruent central angles have congruent
chords.
___
___
−−
4. FE is tangent to O at E.
4. Given.
5. ⬔FEB is a right angle.
5. Definition of a tangent.
6. ⬔FEB and ⬔BED are a linear
pair.
6. Definition of linear pair.
7. ⬔FEB and ⬔BED are
supplementary.
7. Two angles that form a linear pair are
supplementary.
8. ⬔BED is a right angle.
8. A supplement to a right angle is a right
angle.
−−−
9. Diameter BOE
9. Given.
is a semicircle.
10. BCE
10. Definition of a semicircle.
11. ⬔BAE is a right angle.
11. An angle inscribed in a semicircle is always a
right angle.
12. ⬔BAE ⬔BED
12. Right angles are congruent.
13. 䉭ABE 䉭EBD
13. AA.
BD
BE _
14. _
BE
BA
14. Corresponding parts of similar triangles are in
proportion.
5
An appropriate proof with correct conclusion is shown, but one reason is faulty and/or
one statement is missing.
4
A faulty conclusion or no conclusion is drawn from the statements and reasoning.
3
An appropriate proof with correct conclusion is shown, but two reasons are faulty or
two steps are missing.
2
An appropriate proof with correct conclusion is shown, but multiple reasons are faulty
or multiple steps are missing or have errors.
BD , and a reason is given, but no appropriate method is shown.
BE _
_
BE
BA
A zero response is completely incorrect, irrelevant, or incoherent or is a correct response
that was obtained by an obviously incorrect procedure.
1
0
Chapters 1–14
137
2%40!(#
Practice Geometry
Regents Examinations
Practice Regents Examination One
(pages 483–489)
Part I
Allow a total of 56 credits, 2 credits for each of the following correct answers. Allow credit if the student has
written the correct answer instead of the numeral 1, 2, 3, or 4.
1
1 (3) _
23 (4) 4
11 (3) (9, 3)
2
24 (3) right
12 (1) 6
2 (2) 8 √
2
25 (2) 9
13 (4) 16 : 81
1 b2
3
3 (1) _
26 (3) x
14 (3) 18
2
27 (4) I and II only
15 (4) 1
4 (1) 48
1V
16 (2) 26
28 (3) _
3
5 (2) 8 √
8
17 (4) 16
6 (2) 12
18 (2) y
7 (4) r : 2
19 (2) 6 √2
8 (4) 106
20 (1) 86
9 (1) 8
21 (2) 18
10 (3) ⬔AEC is a
22 (2) (0, 1)
right angle.
Part II
For each question, use the specific criteria to award a maximum of 2 credits.
29
138
Score
Explanation
2
2
2
(x 2) (y 4) 2.25, and an appropriate explanation is given.
1
(x 2) 2 (y 4) 2 2.25, but no explanation is given.
0
A zero response is completely incorrect, irrelevant, or incoherent or is a correct response
that was obtained by an obviously incorrect procedure.
30
31
32
33
34
Score
Explanation
2
100, and an appropriate explanation is given.
1
100, but no explanation is given.
0
A zero response is completely incorrect, irrelevant, or incoherent or is a correct response
that was obtained by an obviously incorrect procedure.
Score
Explanation
2
30, and an appropriate explanation is given.
1
30, but no explanation is given.
0
A zero response is completely incorrect, irrelevant, or incoherent or is a correct response
that was obtained by an obviously incorrect procedure.
Score
Explanation
2
3 √
2 , and an appropriate explanation is given.
1
3 √
2 , but no explanation is given.
0
A zero response is completely incorrect, irrelevant, or incoherent or is a correct response
that was obtained by an obviously incorrect procedure.
Score
Explanation
2
16, and an appropriate explanation is given.
1
16, but no explanation is given.
0
A zero response is completely incorrect, irrelevant, or incoherent or is a correct response
that was obtained by an obviously incorrect procedure.
Score
Explanation
2
(1, 0), and an appropriate explanation is given.
1
(1, 0), but no explanation is given.
0
A zero response is completely incorrect, irrelevant, or incoherent or is a correct response
that was obtained by an obviously incorrect procedure.
Part III
For each question, use the specific criteria to award a maximum of 4 credits.
35
Score
Explanation
4
y x 1, and an appropriate explanation is given.
3
An appropriate method is used, but a single computational error is made.
2
An appropriate method is used, but multiple computational errors are made.
1
y x 1, but no explanation is given.
0
A zero response is completely incorrect, irrelevant, or incoherent or is a correct response
that was obtained by an obviously incorrect procedure.
Practice Regents Examination One
139
36
37
Score
Explanation
4
36, and an appropriate explanation is given.
3
An appropriate method is used, but a single computational error is made.
2
An appropriate method is used, but multiple computational errors are made.
1
36, but no explanation is given.
0
A zero response is completely incorrect, irrelevant, or incoherent or is a correct response
that was obtained by an obviously incorrect procedure.
Score
Explanation
4
27, and an appropriate explanation is given.
3
An appropriate method is used, but a single computational error is made.
2
An appropriate method is used, but multiple computational errors are made.
1
27, but no explanation is given.
0
A zero response is completely incorrect, irrelevant, or incoherent or is a correct response
that was obtained by an obviously incorrect procedure.
Part IV
For each question, use the specific criteria to award a maximum of 6 credits.
38
Score
6
140
Explanation
The following proof or another appropriate proof is given.
Statements
−− −−
1. LA 储 EG
Reasons
1. Given.
2. ⬔LAR ⬵ ⬔GER
2. Alternate interior angles are congruent.
3. ⬔ALR ⬵ ⬔EGR
3. Alternate interior angles are congruent.
4. 䉭ALR ⬃ 䉭EGR
RE
AR _
5. _
RL
RG
6. Rhombus PARL
4. AA⬃.
7. AR PA; RL PL
7. Definition of a rhombus.
8. AR RG RL RE
8. The product of means equals the product of extremes.
9. PA RG RE PL
9. Substitution postulate.
5. Corresponding parts of similar triangles are in proportion.
6. Given.
5
An appropriate proof with correct conclusion is shown, but one reason is faulty and/or
one statement is missing.
4
A faulty conclusion or no conclusion is drawn from the statements and reasoning.
3
An appropriate proof with correct conclusion is shown, but three reasons are faulty or
three steps are missing.
2
An appropriate proof with correct conclusion is used, but more than three reasons are
faulty or more than three steps are missing or have errors.
1
PA RG RE PL and a reason is given, but no appropriate method is shown.
0
A zero response is completely incorrect, irrelevant, or incoherent or is a correct response
that was obtained by an obviously incorrect procedure.
Practice Geometry Regents Examinations
Practice Regents Examination Two
(pages 490–496)
Part I
Allow a total of 56 credits, 2 credits for each of the following correct answers. Allow credit if the student has
written the correct answer instead of the numeral 1, 2, 3, or 4.
−− −−
1 (1) EK ⬜ EY
10 (2) 120
21 (2) 5
3
2 (4) 150
11 (2) 20 √3
22 (1) 64 cm
3
12 (3) 24
3 (1) 4 √
23 (4) 4
1
13 (1) 28
24 (3) perpendicular
4 (3) _
2
14 (3) 160
bisectors of the sides
5 (3) 36
15 (3) 2
of the triangle
6 (4) R 90
3
4
_
25
(2) 5 √2
units
16 (1)
7 (4) Use the slope formula
3
26 (4) 36
17 (4) 6 s 2x
to prove diagonals are
27 (1) BC AB
18 (4) 11
perpendicular.
28 (3) equal in area
19 (3) 8
8 (2) 105
29 }
20 (3) 15
9 (4) {2, 5, √
Part II
For each question, use the specific criteria to award a maximum of 2 credits.
29
30
31
Score
Explanation
2
69, and an appropriate explanation is given.
1
69, but no explanation is given.
0
A zero response is completely incorrect, irrelevant, or incoherent or is a correct response
that was obtained by an obviously incorrect procedure.
Score
Explanation
2
(x 4) 2 (y 5) 2 9, and an appropriate explanation is given.
1
(x 4) 2 (y 5) 2 9, but no explanation is given.
0
A zero response is completely incorrect, irrelevant, or incoherent or is a correct response
that was obtained by an obviously incorrect procedure.
Score
Explanation
2
(2x, 3y), and an appropriate explanation is given.
1
(2x, 3y), but no explanation is given.
0
A zero response is completely incorrect, irrelevant, or incoherent or is a correct response
that was obtained by an obviously incorrect procedure.
Practice Regents Examination Two
141
32
33
34
Score
Explanation
2
108, and an appropriate explanation is given.
1
108, but no explanation is given.
0
A zero response is completely incorrect, irrelevant, or incoherent or is a correct response
that was obtained by an obviously incorrect procedure.
Score
Explanation
2
6.8, and an appropriate explanation is given.
1
6.8, but no explanation is given.
0
A zero response is completely incorrect, irrelevant, or incoherent or is a correct response
that was obtained by an obviously incorrect procedure.
Score
Explanation
2
117, and an appropriate explanation is given.
1
117, but no explanation is given.
0
A zero response is completely incorrect, irrelevant, or incoherent or is a correct
response that was obtained by an obviously incorrect procedure.
Part III
For each question, use the specific criteria to award a maximum of 4 credits.
35
36
142
Score
Explanation
4
11.55, and an appropriate explanation is given.
3
An appropriate method is used, but answer is not rounded to the nearest hundredth.
2
An appropriate method is used, but multiple computational errors are made.
1
11.55, but no explanation is given.
0
A zero response is completely incorrect, irrelevant, or incoherent or is a correct response
that was obtained by an obviously incorrect procedure.
Score
Explanation
4
OC 21, m⬔COD 69.5, and an appropriate explanation is given.
3
An appropriate method is used, but a single computational error is made.
2
An appropriate method is used, but only one of the two answers is found.
1
OC 21, m⬔COD 69.5, but no explanation is given.
0
A zero response is completely incorrect, irrelevant, or incoherent or is a correct response
that was obtained by an obviously incorrect procedure.
Practice Geometry Regents Examinations
37
Score
Explanation
4
Secant points of intersection are (0, 5) and (2, 4), and an appropriate explanation is
given.
3
Secant is given, but graph is incorrect.
2
Graph is shown, but secant is not given.
1
Secant is given, but graph is not shown.
0
A zero response is completely incorrect, irrelevant, or incoherent or is a correct response
that was obtained by an obviously incorrect procedure.
Part IV
For each question, use the specific criteria to award a maximum of 6 credits.
38
Score
Explanation
6
Any appropriate method is used to prove that ABCD is a parallelogram, but not a rectangle. For example: showing that the diagonals bisect each other but are not congruent,
or showing that no consecutive sides have negative reciprocal slopes.
5
An appropriate method is used, but a single computational error is made.
4
An appropriate method is used, but two computational errors are made.
3
An appropriate method is used to prove that ABCD is a parallelogram but does not
show that it is not a rectangle.
2
An appropriate method is used to prove that ABCD is a parallelogram, but more than
two computational errors are made and student does not show that it is not a rectangle.
1
An appropriate method is used, but ABCD is not shown to be a parallelogram, and
multiple computational errors are made.
0
A zero response is completely incorrect, irrelevant, or incoherent or is a correct response
that was obtained by an obviously incorrect procedure.
Practice Regents Examination Three
(pages 497–504)
Part 1
Allow a total of 56 credits, 2 credits for each of the following correct answers. Allow credit if the student has
written the correct answer instead of the numeral 1, 2, 3, or 4.
1 (2) 50
22 (1) perpendicular bisectors
11 (2) (x 3) 2 (y 7) 2 2
1
2 (4) 16 : 25
of the sides
12 (3) y _x 4
2
3 (4) (7, 7 √3)
23 (3) 120
13 (3) d 5
2
4 (2) 18 cm
24 (4) obtuse
14 (4) 11
5 (3) {2, 7, 10}
25 (3) 60
15 (1) 60
6 (3) (3, 6.5)
26 (4) 210
3
16 (2) 32 √
3
7 (3) 80 √
27 (1) The perpendicular
17 (2) 16 16 √2
8 (3) ABCD is a parallelobisector of the chord
18 (4) x 1
gram whose diagonals
of a circle bisects the
19 (1) 5
bisect each other.
intercepted arc.
20 (2) y 5x 2
2
_
28
(1)
(3, 5)
9 (2) 3
21 (3) 4 √
3
10 (2) 156
Practice Regents Examination Three
143
Part II
For each question, use the specific criteria to award a maximum of 2 credits.
29
30
31
32
Score
2
216 cubic inches, and an appropriate explanation is given.
1
216 cubic inches, but no explanation is given.
0
A zero response is completely incorrect, irrelevant, or incoherent or is a correct response
that was obtained by an obviously incorrect procedure.
Score
a PA 9, and an appropriate explanation is given.
1
53 and an appropriate explanation is given.
b mCB
0
A zero response is completely incorrect, irrelevant, or incoherent or is a correct response
that was obtained by an obviously incorrect procedure.
Score
16 inches, and an appropriate explanation is given.
1
16 inches, but no explanation is given.
0
A zero response is completely incorrect, irrelevant, or incoherent or is a correct response
that was obtained by an obviously incorrect procedure.
Score
0
144
Explanation
2
1
34
Explanation
1
2
33
Explanation
Score
Explanation
60
_
, and an appropriate explanation is given.
13
60
_
, but no explanation is given.
13
A zero response is completely incorrect, irrelevant, or incoherent or is a correct response
that was obtained by an obviously incorrect procedure.
Explanation
2
32, and an appropriate explanation is given.
1
32, but no explanation is given.
0
A zero response is completely incorrect, irrelevant, or incoherent or is a correct response
that was obtained by an obviously incorrect procedure.
Score
Explanation
2
4, and an appropriate explanation is given.
1
4, but no explanation is given.
0
A zero response is completely incorrect, irrelevant, or incoherent or is a correct response
that was obtained by an obviously incorrect procedure.
Practice Geometry Regents Examinations
Part III
For each question, use the specific criteria to award a maximum of 4 credits.
35
36
37
Score
Explanation
4
16, and an appropriate explanation is given.
3
An appropriate method is used, but a single computational error is made.
2
An appropriate method is used, but incorrect radius is found.
1
16, but no explanation is given.
0
A zero response is completely incorrect, irrelevant, or incoherent or is a correct response
that was obtained by an obviously incorrect procedure.
Score
Explanation
4
40, and an appropriate explanation is given.
3
An appropriate method is used, but a single computational error is made.
2
An appropriate method is used, but multiple computational errors are made.
1
40, but no explanation is given.
0
A zero response is completely incorrect, irrelevant, or incoherent or is a correct response
that was obtained by an obviously incorrect procedure.
Score
4
Explanation
2. Reflexive property of congruence
3. Perpendicular lines meet to form right angles.
4. All right angles are congruent.
5. Two right triangles are similar if an acute angle of one triangle is congruent to an
acute angle of the other.
6. Corresponding sides of similar triangles are in proportion.
7. The product of means equals the product of extremes.
3
One reason is incorrect.
2
Two reasons are incorrect.
1
More than two reasons are incorrect.
0
A zero response is completely incorrect, irrelevant, or incoherent or is a correct response
that was obtained by an obviously incorrect procedure.
Practice Regents Examination Three
145
Part IV
For each question, use the specific criteria to award a maximum of 6 credits.
38
Score
6
Explanation
The following proof or another appropriate proof is given.
Statements
Reasons
1. Parallelogram ABCD,
−−−
with ABG
−− −−− −−− −−−
2. AB 储 CD, AG 储 CD
1. Given.
3. ⬔MGB ⬵ ⬔MDC
3. Alternate interior angles are congruent.
4. ⬔BMG ⬵ ⬔CMD
4. Vertical angles are congruent.
2. Definition of a parallelogram.
−−
5. M is the midpoint of BC.
−−− −−−
6. BM ⬵ MC
7. a 䉭BGM ⬵ 䉭CDM
−− −−−
8. BG ⬵ CD
−− −−−
9. AB ⬵ CD
−− −−
10. b AB ⬵ BG
6. Definition of midpoint.
7. AAS ⬵ AAS.
8. CPCTC.
9. Opposite sides of a parallelogram are congruent.
10. Transitive postulate of congruence
5
An appropriate proof with correct conclusion is shown, but one reason is faulty and/or
one statement is missing.
4
A faulty conclusion or no conclusion is drawn from correct statements and reasoning.
3
An appropriate proof with correct conclusion is shown, but no conclusion is made
−− −−
for AB ⬵ BG.
2
An appropriate proof with correct conclusion is used, but multiple reasons are faulty or
multiple steps are missing or have errors.
−− −−
䉭BGM ⬵ 䉭CDM and AB ⬵ BG, and a reason is given, but no appropriate proof is
shown.
1
0
146
5. Given.
A zero response is completely incorrect, irrelevant, or incoherent or is a correct response
that was obtained by an obviously incorrect procedure.
Practice Geometry Regents Examinations