Download Geo wo Obj Sheets 1-50

Geometry Course Outline
Primary Textbook: Ron Larson, Laurie Boswell, Timothy D. Kanold, and Lee Stiff
Geometry, McDougal Littell, 2007
Unit 1: Essentials of Geometry

Describing geometric figures

Measuring geometric figures

Understanding equality and congruence
Unit 2: Reasoning and Proof

Use inductive and deductive reasoning

Understanding geometric relationships in diagrams

Writing proofs of geometric relationships
Unit 3: Parallel and Perpendicular Lines

Using properties of parallel and perpendicular lines

Proving relationships using angle measures

Making connections to lines in algebra
Unit 4: Congruent Triangles

Classifying triangles by sides and angles

Proving that triangles are congruent

Using coordinate geometry to investigate triangle relationships
Unit 5: Relationships within Triangles

Using properties of special segments in triangles

Using triangle inequalities to determine what triangles are possible

Extending methods for justifying and proving relationships
Unit 6: Similarity

Using ratios and proportions to solve geometry problems

Showing that triangles are similar

Using indirect measurement and similarity
Unit 7: Right Triangles and Trigonometry

Using the Pythagorean Theorem and its converse

Using special relationships in right triangles

Using trigonometric ratios to solve right triangles
Unit 8: Quadrilaterals

Using angle relationships in polygons

Using properties of parallelograms

Classifying quadrilaterals by their properties
(cont)
Geometry
-1-
Geometry Course Outline
(cont)
Unit 9: Properties of Transformations

Performing congruence and similarity transformations

Making real-world connections to symmetry and tessellations

Applying matrices and vectors in Geometry
Unit 10: Properties of Circles

Using properties of segments that intersect circles

Applying angle relationships in circles

Using circles in the coordinate plane
Unit 11: Measuring Length and Area

Using area formulas for polygons

Relating length, perimeter, and area ratios in similar polygons

Comparing measures for parts of circles and the whole circle
Unit 12: Surface Area and Volume of Solids

Exploring solids and their properties

Solving problems using surface area and volume

Connecting similarity to solids
Geometry
-2-
Geometry Unit 1
Pre-Rev. A
Review linear equations and proportions.
Pre-Rev. B
Review simplifying radicals and systems of equations.
1.
Identify points, lines and planes. (Section 1.1)
2.
Use segments and congruence. (Section 1.2)
3.
Use midpoint and distance formulas. (Section 1.3)
4.
Measure and classify angles. (Section 1.4)
5.
Describe angle pair relationships. (Section 1.5)
Review
Geometry
-3-
Unit 1 Pre-Review A
Algebra Review
Solve the following equations.
1.
n–4=9
2.
p + 7 = –7
3.
8 + f = –8
4.
4c = 96
5.
= 21
6.
–
7.
5n + 4 = 29
8.
8y – 7 = 17
9.
9x – 5 = –14
10.
15 + 5y = 20
11.
5y – 3 – 4y = 5
12.
m + 3m + 2 + 2m = 14
13.
5x + 4 – 2x = 2 + 2
14.
3x – 1 = 8x + 9
15.
4y = 2y + 6
16.
4(x + 5) = 28
17.
5(x – 3) + 8 = 18
18.
3(5x – 4) = 8x + 2
Solve the following proportions.
19.
20.
21.
22.
23.
24.
Geometry Unit 1 Pre-Review B
Radical and Systems of Equations Review
Simplify:
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
12.
Solve the following for x and y:
13.
y = 3x
14.
5x + y = 24
16.
3x + 16 = 100
2x = 7y
Geometry
17.
4x – 5y = 92
x = 7y
15.
x = 8 + 3y
2x – 5y = 8
6x + 12 = 180
3x = 4y
18.
4x = 10y
7x + 90 = 125
-4-
= 6
Worksheet 5A
1.
A and B are complementary.
A and
If m C = 100°, determine the measures of
2.
L and M are complementary.
M and P are supplementary.
If m L = 20°, determine the measures of M and P.
3.
C are supplementary.
A and B.
A and B are complementary.
A and C are supplementary.
A
B. Determine the measures of A, B and
C.
4.
The larger of two supplementary angles measures 8 times the smaller. Determine the
measures of the two angles.
5.
The measure of an angle is 4 times the measure of its complement. Determine the
measures of both angles.
Unit 1 Worksheet 5B
1.
The complement of an angle is five times the measure of the angle itself. Determine the
angle and its complement.
2.
Determine the measure of an angle that is 50° more than that of its complement.
3.
Determine the measure of an angle that is 60° more than that of its supplement.
4.
The supplement of an angle is 30° less than twice the measure of the angle itself.
Determine the angle and its supplement.
5.
The supplement of an angle is twice as large as the angle itself. Determine the angle and
its supplement.
The complement of an angle is 6° less than twice the measure of the angle itself.
Determine the angle and its complement.
Two angles are congruent and complementary. Determine their measures.
6.
7.
8.
9.
10.
The complement of an angle is twice as large as the angle itself. Determine the angle and
its complement.
The supplement of an angle is 20° more than three times the angle itself. Determine the
angle and its supplement.
Determine the measure of an angle that is 18° more than half of its complement.
Geometry
-5-
Unit 1 Review
In problems 1 – 19 select the correct multiple choice response.
Diagrams are not drawn to scale.
1.
Write a name for the given figure
a.
2.
3.
4.
b.
A
20
c.
d.
fe
Which of the following is not a name for the given figure?
et
h
a.
b.
B
A
l
c.
d.
Name the vertex of
SRT
a.
S
b.
R
c.
T
d.
cannot determine without a diagram
h
Classify the given angle
a.
right angle
b.
obtuse angle
c.
acute angle
d.
straight angle
5.
6.
E
1 and
2 are what kind of angles?
a.
vertical angles
b.
congruent angles
c.
complementary angles
d.
adjacent angles
2
1
Which statement is not true regarding the two angles?
a.
A
B
b.
A and
B are complementary angles
c.
A and
B are vertical angles
d.
A and
B are acute angles
45°
B
7.
All of the following are names for the angle shown except which one?
E
a.
EFM
b.
F
F
c.
MFE
d.
FEM
8.
Which angles are vertical angles?
M
a.
2 and
4
b.
1 and
5
c.
2 and
5
d.
1 and
3
2
1
5
(cont)
Geometry
45°
A
-6-
4
3
Review (cont)
9.
In the figure,
m
bisects
2 = (9x)°, find m
PQR. If m
1 = (6x + 18)° and
S
PQR
a.
88°
b.
105°
d.
110°
e.
135°
c.
P
108°
1
2
Q
R
10.
11.
12.
13.
bisects
measure of
EXG.
EXG?
If m
EXF = 34°, what is the
a.
34°
b.
68°
c.
56°
d.
45°
E
F
X
What is another name for angle 2?
a.
U
b.
TUM
c.
UKM
d.
MUK
K
2
1
T
In the figure shown, m 1 = (5x)° m 2 = (6x + 10)°
Which equation could be used to find the value of x?
a.
5x = 6x + 10
b.
5x = 120
c.
6x + 10 = 120
d.
5x + 6x + 10 = 120
G
M
U
m
ABC = 120°
C
2
1
T
In the figure name three collinear points.
a.
S, P, Y
b.
S, P, T
c.
T, P, X
d.
T, X, Y
X
B
A
Y
P
S
14.
Which angle is a supplement to TSY?
a.
PST
b.
YSX
c.
PSX
d.
P
L
PSY
X
S
T
(cont)
Geometry
-7-
Y
Review (cont)
15.
bisects ZRL
ZRB = (4x + 16)°
LRB = 88°
L
B
Which equation could be used to find the value of x?
a.
b.
c.
d.
16.
4x + 16 = 88
4x + 16 + 88 = 90
2(4x + 16) = 88
x = 88 + 4x + 16
18.
19.
R
Which of the following statements is true for the figure shown and using the
segment addition postulate.
a.
b.
c.
17.
Z
DB = BA
DA = DB + BA
B is the midpoint of DA
D
In the figure shown, B is the midpoint of
DB = 5x – 1,
BA = 4x + 6
DA = 68
Identify all equations below that are true.
D
a.
5x – 1 = 4x + 6
b.
5x – 1 + 4x + 6 = 68
c.
2(5x – 1) = 68
d.
2(4x + 6) = 68
The endpoints of
midpoint M.
a.
(–4, 12)
B
A
B
A
are R (5, –1) and S (–9, 13). Find the coordinates of the
b.
(–2, 6)
c.
(–7, 7)
d.
(7, 6)
Which of the following is the distance formula?
a.
b.
c.
d.
For problems 20 – 32 answer True or False:
20.
21.
ABC is a right angle
Point B is between points A and D
C
22.
23.
ABC
A
DBC
24.
Points A, B and D are collinear
25.
Point C is between points A and D
D
Use this figure for
problems 20 - 25
(cont)
Geometry
B
-8-
Review (cont)
26.
If a straight angle is bisected the resulting angles will always be right angles
27.
If a right angle is bisected the resulting angles will always be acute angles.
28.
If an obtuse angle is bisected the resulting angles will always be obtuse angles.
29.
Point B is between points P and S
30.
If two angles are obtuse, then they are congruent.
31.
If two angles are right angles, then they are congruent.
32.
If two angles are right angles, then they are adjacent.
P
B
S
Solve the following problems showing all work.
33.
A and B are complementary.
A and C are supplementary.
If m C is 140°, determine the measures of A and B
Geometry
-9-
Unit 2 Objective 0
(Note: Diagrams are not drawn to scale.)
• A polygon is a closed plane figure with the following properties:
It is formed by three or more line segments called ____________.
Each segment intersects exactly ____ other segments, one at each endpoint.
No two segments with a common ________________are collinear.
1.
Draw 3 polygons.
2.
Draw 3 figures that are not polygons.
3.
This polygon is a convex polygon.
Draw three more.
4.
This polygon is a concave polygon (not convex). Draw three more.
•
Polygons are classified according to the number of sides they have. Name each.
3 sides ____________________
8 sides ____________________
4 sides ____________________
9 sides ____________________
5 sides ____________________
10 sides ___________________
6 sides ____________________
12 sides ___________________
7 sides ____________________
n sides ___________________
•
In an _____________________ polygon, all sides are equal.
•
In an _____________ polygon, all angles in the interior of the polygon are congruent.
•
A convex polygon that is both equilateral and equiangular is called a ________polygon.
Classify each polygon by the number of sides. Tell whether the polygon is equilateral, equiangular,
regular, or none of these
5. ____________
6. ____________ 7. _____________ 8. _____________
(cont)
Geometry
- 10 -
Unit 2 Objective 0 (cont)
For each regular polygon find the value of x, the length of a side and the perimeter of the polygon.
9.
x = _______
Length of each side _______
10.
Perimeter = _______
x = _______
Length of each side _______
Perimeter = _______
5x – 27
4x + 3
5x – 1
2x – 6
Use the coordinate grid for problems 11 and 12. Show all work!
11.
Triangle QRS has vertices Q (1,2), R (4, 6), and
S (5,2). What is the perimeter of triangle QRS?
(Draw triangle QRS. Find the side lengths using the
distance formula, then find the perimeter.)
SR = _________________ QR = _________________
QS = _________________ Perimeter = ____________
12.
Quadrilateral MATH has vertices M (–4, –2), A (–1, 1), T (2, –2) and H (–1, –5).
What is the perimeter of quadrilateral MATH? (Show your work. Use the distance formula)
MA = _________________ AT = _________________
TH = _________________ HM = _________________
Perimeter = ____________
Is quad MATH equilateral? ________ Why or why not?_________
What additional information would be needed to say quad MATH is regular?
_____________________________________________________________
What is another polygon name for quad MATH if it is regular?_____
Geometry
- 11 -
Geometry Unit 2
1.
Use inductive and deductive reasoning. Give counterexamples to disprove a
statement. (Section 2.1, Section 2.3)
2.
Analyze conditional statements. (Section 2.2)
3.
Use postulates and diagrams. (Section 2.4)
4.
Reason using properties from Algebra. (Section 2.5)
5.
Prove statements about segments and angles. (Section 2.6)
6.
Prove angle pair relationships. (Section 2.7)
Review
Geometry
- 12 -
Unit 2 Worksheet 1
Inductive
Reasoning
When you reason that what has happened before will happen again, without exception, you are
using inductive reasoning. Inductive reasoning consists of observing data, recognizing patterns
and making decisions based on past experiences. Through observations we are lead to make a
conjecture which may be true or false.
Deductive
Reasoning
“Deduce” means to reason from known facts. When you reason deductively, you reach a
conclusion using established rules. You start with statements that are considered true and then
show that other statements follow from them.
Inductive Reasoning
Deductive Reasoning
uses past observations
uses patterns
uses unproven statements
uses facts
uses definitions
uses postulates
uses corollaries
uses previous theorems
CAN BE USED IN
MATHEMATICAL PROOFS
CANNOT BE USED IN
MATHEMATICAL PROOFS
Inductive
Reasoning -
can be Inaccurate because it’s based
on feelings, observations, patterns
In problems 1 – 10 state whether the reasoning represents deductive or inductive reasoning.
1.
Conclusions are based on feelings.
2.
Conclusions are based on observing objects.
3.
Conclusions are based on definitions.
4.
Conclusions are based on proven facts and accepted statements.
5.
Conclusions are based on previous patterns.
6.
Conclusions are based on suspicions.
7.
Conclusions are based on other theorems.
8.
Conclusions are based on established laws.
9.
The definition of an even number is that it is a number that when divided by 2 has
a remainder of 0. When Sue divided 16 by 2 she got a remainder of 0. Sue
conjectured that 16 is an even number. What type of reasoning did she use?
(cont)
Geometry
- 13 -
Unit 2 Worksheet 1 (cont)
10.
John was told to fill in the sequence 10, 20, 30, ___.
He conjectured that the missing term was 50. What type of reasoning did he use?
10, 20,
30,
50
add
In problems 11 – 14, select the correct multiple choice response:
11.
Which number serves as a counterexample to the statement?
The square of every integer is an even number.
a. 2
b. 6
c. 5
d. 10
12.
13.
The table shows an expression evaluated for four different values of x.
x
2x + 5
-2
1
0
5
1
7
4
13
a. –10
b. –2
c. –1
d. 0
Which number serves as a counterexample to the statement
All rational numbers can be written as terminating decimals.
a.
14.
Rick concluded that for every x the value of 2x + 5
produces a positive number. Which value of x
serves as a counterexample to prove Rick’s conclusion
false?
= 0.5
b.
= 1.75
c.
= 0.
d.
= 0.018
Which multiple serves as a counterexample to the statement
If two integers are added together and the sum is even, then the original two
integers are even.
a.
2, 4
Geometry
b.
1, 3
c.
- 14 -
0, 6
d.
–2, – 4
Unit 2 Worksheet 2
NOTE: Diagrams are not drawn to scale.
Decide whether the statement is true or false. If false, provide a counterexample.
1.
If it is a weekend day, then it is Saturday.
2.
If an angle is acute, then its measure is less than 90°.
3.
If
4.
If a = b, then a + c = b + c
5.
If a figure is a rectangle, then it has 4 sides.
6.
If n > 5, then n > 7.
, then
Write the converse for the statements below and determine if the converse is true or false. If
false, provide a counterexample.
7.
If I have 2 dimes and 1 nickel, then I have 25 cents.
8.
If
= 90°, then
is a right angle.
2
9.
If x = – 6, then x = 36.
10.
If you can divide a number by 4, then you can divide the number by 2.
Select the correct multiple choice response:
11.
If two angles share a common vertex, then they are adjacent
Which of the following serves as a counterexample to the assertion above?
a.
b.
1
2
1
c.
d.
1
2
1
(cont)
Geometry
2
- 15 -
2
Unit 2 Worksheet 2 (cont)
12.
If two lines are coplanar, then they intersect.
Which of the following serves as a counterexample to the assertion above?
a.
b.
c.
p
m
13.
A pair of supplementary angles are adjacent to each other.
Which of the following serves as a counterexample to the assertion above?
a.
b.
c.
60°
40°
140°
90°
90°
120°
14.
15.
The definition of congruent segments is:
If two line segments have the same length then they are congruent segments.
a.
Write the converse of this definition
b.
Write the definition as a biconditional
The definition of perpendicular lines is:
If two lines intersect to form a right angle, then they are perpendicular lines.
a.
Write the converse of this definition
b.
Write the definition as a biconditional
Geometry
- 16 -
Unit 2 Worksheet 4
NOTE: Diagrams are not drawn to scale.
Name the property illustrated below:
1.
2.
If
3.
If RS = TW, then TW = RS
4.
If x + 5 = 16, then x = 11
5.
If 5y = – 20,
6.
2(a + b) = 2a + 2b
7.
If 2x + y = 70 and y = 3x, then 2x + 3x = 70
8.
9.
If AB = CD and CD = 23, then AB = 23
Justify each step:
2x + 3 = 11
Given
a.
2x = 8
______________________
b.
x = 4
______________________
Justify each step:
10.
and
,
then y = – 4
x = 6 + 2x
Given
a.
b.
3x = 4(6 + 2x)
3x = 24 + 8x
______________________
______________________
c.
– 5x = 24
______________________
d.
11.
then
x = –
______________________
A
Justify each step:
Given:
1
Prove:
O
Statements
B
C
2
3
D
Reasons
1.
1. ______________________
2.
2. ______________________
3.
=
3. ______________________
4.
4. ______________________
5.
5. ______________________
(cont)
Geometry
- 17 -
Unit 2 Worksheet 4 (cont)
12.
Given:
FL = AT
Prove:
FA = LT
F
L
A
Statements
Reasons
1.
FL = AT
1. ______________________
2.
LA = LA
2. ______________________
3.
3. ______________________
4.
4. ______________________
5.
FA = LT
5. ______________________
R
13.
Given:
and
intersect at S so that
RS = PS and ST = SQ
Prove:
T
P
S
RT = PQ
Q
Statements
Reasons
1.
RS = PS and ST = SQ
1. ______________________
2.
ST = ST
2. ______________________
3.
3. ______________________
4.
4. ______________________
5.
5. ______________________
6.
RT = PQ
6. ______________________
(cont)
Geometry
- 18 -
T
Unit 2 Worksheet 4 (cont)
14.
Given:
DW = ON
Prove:
DO = WN
D
O
W
1.
Reasons
DW = ON
1. ______________________
2.
2. ______________________
3.
=
3. ______________________
4.
5.
15.
N
Statements
Given:
4. ______________________
DO = WN
5. ______________________
and
Prove:
3
R
Statements
1.
Q
P
1
Reasons
and
1. ______________________
2.
2.______________________
3.
3. ______________________
4.
4. ______________________
5.
5. ______________________
6.
6. ______________________
Geometry
- 19 -
2
4
T
Unit 2
Simple Proofs Worksheet 5A
Identify the property used. Choose from this list of justifications.
Addition, Subtraction, Multiplication, Division, Substitution, Reflexive, Symmetric, Transitive
Statement
Reason
3a = 6
a=2
Given
2.
a+2=5
5=a+2
Given
3.
CD + DE = CE
CD = CE – DE
4.
5.
1.
Statement
Reason
b=3
3=b
Given
.13.
AB = 7
AB + 2 = 9
Given
Given
14.
a=3
a–5=–2
Given
b=5
3b = 15
Given
15.
AB = 10
2AB = 20
Given
2y = 50
Given
16.
xy = 10
Given
12.
222
y = 25
6.
AB = CD
AB + 5 = CD + 5
7.
AB = 3 and XY = 3
AB = XY
8.
3QX = 10
=5
Given
17.
XY + YZ = 15 and 15 = WV
XY + YZ = WV
Given
18.
RO = NP
3RO = 3NP
Given
Given
19.
15 – AB = DE
15 = DE + AB
Given
QX =
9.
AB + CD = 30 and CD = 10
AB + 10 = 30
Given
20.
AB + 10 = 30
AB = 20
Given
10.
Given
21.
a < b and b < c
a<c
Given
0.
AB = 5 and CD = AB
CD = 5
11.
a + b > c and c = 10
Given
22.
a<b
Given
a + b > 10
Geometry
2a < 2b
- 20 -
Unit 2 Worksheet 5B
NOTE: Diagrams are not drawn to scale.
In problems 1 – 14 name the property illustrated.
1.
If m = k and k = d, then m = d
2.
3.
If 5 = 4 + 1, then 4 + 1 = 5
4.
If Y is between X and Z, then XY + YZ = XZ
5.
If x + 7 = 9, then x = 2
6.
If point A is in the interior of
.
X
Y
Z
, then
X
A
Y
Z
7.
a(b + c) = ab + ac
8.
If
9.
If
m+d=7
and d = k,
then m + k = 7
= 90°, then
10.
and
are complements.
= 180°
2
1
11.
If y – 5 = 11, then y = 16
12.
If P is between A and B and AP = PB. then P is the midpoint of AB
13.
If
14.
If WY = KD and KD = AB, then WY = AB
15.
If B is between A and C
16.
Write a counterexample to the statement:
A
P
X
, then
bisects
Y
and AB = BC, what do you call point B?
If xy = 6, then x = 2 and y = 3.
Geometry
W
- 21 -
Z
B
Unit 2 Worksheet 6
NOTE: Diagrams are not drawn to scale.
1.
H
Given:
1
Prove:
D
Statements
2.
1.
2.
2.
3.
3.
4.
4.
5.
5.
KP = ST
KR = SV
PR = TV
Statements
3
E
F
K
P
R
S
T
V
Reasons
1.
3.
2
Reasons
1.
Given:
Prove:
G
1.
2.
PR = PR
2.
3.
KP + PR = ST + PR
3.
4.
KP + PR = ST + TV
4.
5.
KP + PR = KR
ST + TV = SV
5.
6.
KR = SV
6.
Given:
1 2
Prove:
Statements
Reasons
1.
1.
2.
2.
3.
4.
180° = 180°
3.
4.
5.
5.
6.
6.
(cont)
Geometry
- 22 -
3 4
Unit 2 Worksheet 6 (cont)
4.
A
Given:
1
Prove:
E
2
3
I
U
O
Statements
5.
Reasons
1.
1.
2.
2.
3.
3.
4.
4.
5.
5.
Given:
M is the midpoint of
N is the midpoint of
PQ = RS
Prove:
PM = RN
Statements
1.
P
M
Q
R
N
S
Reasons
M is the midpoint of
1.
N is the midpoint of
2.
PM =
2.
RN =
3.
PQ = RS
3.
4.
=
4.
5.
PM = RN
5.
(cont)
Geometry
- 23 -
Given
Unit 2 Worksheet 6 (cont)
S
6.
Given:
1
bisects
bisects
U
Statements
1.
Reasons
bisects
bisects
V
T
T
Prove:
7.
R
2
1.
2.
2.
3.
3.
4.
4.
5.
5.
Given
A
Given:
is complementary to
2
Prove:
1
3
C
Statements
Reasons
1.
1.
2.
is a right angle
2.
3.
3.
4.
4.
5.
5.
6.
and
are complementary
6.
7.
and
are complementary
7.
8.
8.
9.
10.
11.
9.
10.
11.
(cont)
Geometry
D
- 24 -
Given
B
Unit 2 Worksheet 6 (cont)
8.
Given:
D
A
bisects
Prove:
B
4
1
2
E
3
C
Statements
1.
2.
3.
4.
9.
bisects
___
Reasons
1.
2.
3.
4.
___
Definition angle bisector
Vertical angles are
Given:
Prove:
is supplementary to
1
2 4
5 3
Statements
1.
1.
2.
2.
3.
3.
4.
4.
5.
5.
6.
10.
Reasons
and
are supplementary
Given: AC = BD
Prove: AB = CD
1.
2.
3.
4.
5.
6.
A
Statements
AC = BD
Reasons
1.
AB + BC = AC
BC + CD = BD
AB + BC = BC + CD
BC = BC
AB = CD
2.
3.
4.
5.
(cont)
Geometry
B C
- 25 -
D
P
Unit 2 Worksheet 6 (cont)
11.
Given:
3
2
1
X
Prove:
Statements
Reasons
1.
2.
3.
4.
1.
2.
3.
4.
5.
5.
Given:
,
R
S
B
A
12.
Q
1
2
C
E
D
3
4
Prove:
X
Statements
1.
13.
Reasons
,
1.
2.
2.
3.
3.
4.
4.
5.
5.
6.
6.
Given: AC = DF, AB = DE
Prove: BC = EF
Statements
Reasons
1.
AC = DF
1.
2.
AC = AB + BC
DF = DE + EF
2.
3.
AB + BC = DE + EF
3.
4.
5.
6.
7.
AB = DE
AB + BC = AB + EF
AB = AB
BC = EF
4.
5.
6.
7.
(cont)
Geometry
Y
- 26 -
Given
A
B
C
D
E
F
F
Unit 2 Worksheet 6 (cont)
14.
Given:
= 15
Prove:
x=9
Statements
Reasons
1.
15.
1.
2.
5x = 45
2.
3.
x=9
3.
T
Given:
X
1
Prove:
U
Statements
Reasons
1.
1.
2.
2.
3.
3.
4.
4.
5.
5.
6.
6.
7.
7.
Geometry
- 27 -
Given
4
2
V
3
W
Unit 2 Review
In problems 1 - 13 select the correct multiple choice response.
NOTE: Diagrams are not drawn to scale.
1.
Which of the following is a true statement?
a.
Parallel lines always intersect.
b.
Intersecting lines are never parallel
c.
Perpendicular lines never intersect.
d.
Intersecting lines are always perpendicular.
2.
In the figure,
= 80°.
What is the measure of
?
a.
40°
b.
180°
c.
220°
d.
140°
is bisected by
C
.
B
D
A
E
N
3.
4.
= 20°,
Find
a.
c.
If
a.
b.
c.
d.
5.
M
20°
90°
L
b.
160°
d.
110°
, which of the following must be true?
K
P
R
B
B is the midpoint of
A is the midpoint of
A is the midpoint of
R
The intersection of Plane R with Plane P is
a.
Point K
c.
T
A
b.
T
A
d.
P
W
e.
K
M
C
F
6.
Which of the following is a counterexample to the
statement
All real numbers have a reciprocal.
a.
5
b.
0
c.
7.
1
d.
–7
Which of the following is the name for reasoning that uses facts, definitions,
accepted properties and theorems.
a.
Logical reasoning
b.
Inductive reasoning
c.
Deductive reasoning
d.
Observational reasoning
(cont)
Geometry
- 28 -
Unit 2 Review (cont)
8.
Identify the hypothesis in the following statement:
If the sum of two angles is 90°, then the angles are complementary.
a.
There are two angles
b.
If the sum of two angles is 90°
c.
Then the angles are complementary
d.
Angles are complementary if their sum is 90°
9.
Which diagram shows two angles that are supplementary and adjacent?
a.
b.
120°
30°
30°
c.
d.
140°
10.
135°
45°
30°
40°
150°
The pair of angles
and
a.
supplementary angles
b.
vertical angles
c.
adjacent angles
d.
right angles
can best be classified as _____
A
11.
Linear angles are always ______
a.
obtuse
b.
complementary
c.
supplementary
d.
right
12.
Linear angles are NEVER _______
a.
vertical
b.
adjacent
c.
congruent
d.
supplementary
13.
If two angles are complementary, then the sum of their degree measures is
a.
45°
b.
60°
c.
90°
d.
180°
(cont)
Geometry
- 29 -
C
B
O
Unit 2 Review (cont)
In problems 14 - 21 name the property or definition illustrated.
14.
If 6x – 7 = 29, then 6x = 36
15.
3(x + y) = 3x + 3y
16.
If
17.
AC = AC
18.
If
19.
If M is the midpoint of AB, then AM = MB
20.
If
21.
Given: x + y = 14 and x = 5, then
and
then
is a right angle, then
= 90°
= 90°, then
and
are complementary
B
5 + y = 14
C
Use this diagram to answer questions 22 – 24.
E
22.
If
= 37°, calculate
23.
If
24.
If E is the midpoint of AC and AC = 28, calculate AE
25.
Determine the value of x and
= 88°, calculate
A
Determine the value of x,
D
C
A
B
26.
3
and
(10x – 8)°
(6x + 2)°
D
A
and
(4x + 12)°
27.
B
E
(6x – 14)°
D
Use the diagram to find the values of x and y
(7x + 12)°
(3x + 28)°
(4y – 2)°
(4y + 38)°
28.
The complement of an angle is three more than twice the measure of the angle
itself. Find the measure of the angle and the complement.
Write the following definitions as biconditionals.
29.
If points are collinear, then they all lie in one line.
30.
If points lie in one plane, then they are coplanar
(cont)
Geometry
- 30 -
C
31.
Unit 2 Review (cont)
Complete the proof by filling in the reasons.
Given:
6(x – 4) = x + 16
Prove:
x=8
Statements
32.
Reasons
1.
6(x – 4) = x + 16
1.___________________
2.
6x – 24 = x + 16
2.___________________
3.
5x – 24 = 16
3.___________________
4.
5x = 40
4.___________________
5.
x=8
5.___________________
Justify the reasons on the following proof:
Given:
Prove:
MO = LD
ML = OD
M
1.
Statements
MO = LD
Reasons
1.___________________
2.
OL = OL
2.___________________
3.
MO + OL = LD + OL
3.___________________
4.
MO + OL = ML
LD + OL = OD
4.___________________
5.
ML = OD
5.___________________
D
L
O
Use the diagram for problems 33 - 38.
= 90°,
33.
Calculate
34.
Calculate
35.
Calculate
36.
Calculate
37.
Calculate
38.
Calculate
39.
Calculate
40.
Calculate
Geometry
= 40°,
= 60°
C
B
A
- 31 -
E
O
H
.
D
G
F
Unit 2 Geometry Properties Review
Directions: Match the name of the properties in the left column with the definitions in the right column.
1.
Segment Addition Postulate (pg 10)
a)
Two angles whose sum is 180°
2.
3.
Definition Midpoint (pg 15)
Angle Addition Postulate (pg 25)
b)
c)
a=a
If a = b, then a can be replaced with b
4.
5.
Definition Right Angle (pg 25)
Congruent Angles (pg 26)
d)
e)
Two angles whose sum is 90°
Two adjacent angles that are supplementary
6.
Angle Bisector (pg 28)
f)
Two angles that share a common vertex and
side, but have no common interior points
7.
Defn. Complementary Angles (pg 35)
g)
If B is between A and C, then AB + BC = AC
8.
Defn. Supplementary Angles (pg 35)
h)
If a = b, then a – c = b – c
9.
Adjacent Angles (pg 35)
i)
Two angles whose sides form two pairs of
opposite rays. The angles are congruent.
10.
Definition of Perpendicular Lines (pg
j)
If a = b, then
81)
=
, (
0)
11.
Addition Property of Equality (pg 105)
k)
A ray that divides an angle into two congruent
angles
12.
Subtraction Prop.of Equality (pg 105)
l)
Two lines that form a right angle
13.
Mult. Prop. of Equality (pg 105)
m)
If a = b, then b = a
14.
Division Property of Equality (pg 105)
n)
An angle whose measure is 90°
15.
Substitution Prop. of Equality (pg 105)
o)
If a = b, then a c = b c
16.
Distributive Property (pg 106)
p)
Two angles that have the same measure
17.
Reflexive Property of Equality (pg 107)
q)
If P is in the interior of
18.
Symmetric Prop. of Equality (pg 107)
r)
a (b + c) = ab + ac
19.
Transitive Property of Equality (pg 107)
s)
M is on AB and AM = MB,
20.
Linear Pair Postulate (pg 126)
t)
21.
Vert. Angles Congruence Thm. (pg 126)
If a = b, then a + c = b + c
u) If a = b and b = c, then a = c
Geometry
- 32 -
, then
Unit 2 Complements and Supplements
Set up an equation for each problem, then solve for x. Use your answer for ‘x’ to determine the
angle measures for the problem.
1.
The complement of an angle is five times the measure of the angle itself. Determine
the angle and its complement.
2.
The supplement of an angle is 30° less than twice the measure of the angle itself.
Determine the angle and its supplement.
3.
The supplement of an angle is twice as large as the angle itself. Determine the angle
and its supplement.
4.
The complement of an angle is 6° less than twice the measure of the angle itself.
Determine the angle and its complement.
5.
Three times the measure of the supplement of an angle is equal to eight times
the measure of its complement. Determine the angle, its complement, and its
supplement.
6.
Two angles are congruent and complementary. Determine their measures.
7.
Two angles are congruent and supplementary. Determine their measures.
8.
The complement of an angle is twice as large as the angle itself. Determine the angle
and its complement.
9.
The complement of an angle is 10° less than the angle itself. Determine the angle and
its complement.
10.
The supplement of an angle is 20° more than three times the angle itself. Determine
the angle and its supplement.
Geometry
- 33 -
Unit 2 Bisectors, Midpoints, Complements, Supplements, Vertical Angles
1.
bisects
= 2x + 20
a.
b.
c.
2.
3.
1 2
D
3
4
O
= 2x + 5
1
O
Determine the value of x
Determine
Determine
E
M
2
3
4
N
P
R
U
= 12x – 4
Determine the value of x
Determine
Determine
W
1
0
4
3
V
2
Y
Z
4.
C
L
bisects
= 5x + 2
a.
b.
c.
B
A
Calculate the value of x
Calculate
Calculate
bisects
= 3x – 7
a.
b.
c.
= 5x + 5
P
M is the midpoint of LN
LM = 4 + 3x
MN = 7
Calculate the value of x
O
M
N
L
5.
6.
C is the midpoint of BD
BC = 3x – 4 CD = 17
Find the value of x
J is the midpoint of HK
HK = 40 JK = 2x + 8
Find the value of x
7.
= 18x + 6
a.
Find the value of x
b.
Find
c.
Find
A
B
F G
E
F
E
- 34 -
D
J K
H
= 10x
(cont)
Geometry
C
D
G
H
Unit 2 Bisectors, Midpoints, Complements, Supplements, Vertical Angles (cont)
8.
Are
and
complementary? Explain why or why not.
X
47°
43°
Y
9.
Are
and
complementary? Explain why or why not.
1
10.
Are
and
2
complementary? Explain why or why not.
M
37°
53°
P
11.
Calculate the supplement of
12.
= 140°,
if
N
= 35°
Determine the measures of the remaining angles.
a.
B
A
b.
E
c.
C
D
13.
Two complementary angles are congruent. Determine their measures. Show algebraic
work.
14.
Two supplementary angles are congruent. Determine their measures. Show algebraic
work.
In the diagram
is a right angle.
15.
Name another right angle.
A
16.
Name two complementary angles.
17.
Name two congruent supplementary angles.
18.
Name two non-congruent supplementary angles.
19.
Name two acute vertical angles.
20.
Name two obtuse vertical angles.
Geometry
B
F
C
(cont)
- 35 -
E
D
Unit 2 Bisectors, Midpoints, Complements, Supplements, Vertical Angles (cont)
In the diagram,
= 35°,
bisects
,
U
= 120°.
V
T
S
O
Find the measure of each angle below.
21.
22.
23.
24.
25.
26.
W
X
Z
Y
Determine the value of x. Show algebraic work.
27.
28.
29.
(3x–5)°
36°
(3x+8)°
70°
Geometry
(6x–22)°
- 36 -
64°
4x°
Unit 2 Angle Pairs
Use the diagram to decide whether the statement is true or false.
1.
If
= 47°, then
= 43°
2.
If
= 47°, then
= 47°
2
1
3.
3
4
4.
Make a sketch of the given information. Label all angles which can be determined.
5. Adjacent complementary angles
6. Nonadjacent supplementary angles
where one angle measures 42°
where one angle measures 42°
7. Congruent linear pairs
8. Vertical angles which measure 42°
9.
10.
and
are adjacent
complementary angles.
and
are adjacent
complementary angles.
Determine the value of x and y .
(13x + 9)°
12.
A
2(3y – 25)°
E
(4y + 2)° (15x – 1)°
B
are complementary
are complementary
are vertical angles.
Calculate the measure of each angle in the diagram.
A
11.
and
and
and
D
(4x + 10)°
B
C
13x°
E
2(y + 25)° (2y – 30)°
D
C
A
13.
14.
4y° (17y – 9)°
B
E
(21x – 3)° (5x + 1)°
7x° 13y
D
B
C
Geometry
A
E
(16y – 27)° (5x + 18)°
C
- 37 -
D
Unit 2 Special Pairs of Angles
Determine the measures of a complement and a supplement of
complement supplement
1.
2.
3.
4.
complement supplement
= 36°
= 70°
= 49.2°
= 11°
5.
6.
7.
8.
= 5°
= 29°
= x°
= 2x°
In the diagram,
= 90°
9.
is complementary to ________.
10.
is supplementary to ________.
11.
is adjacent to angle ________.
12.
If
= 80° and
V
= 50°
= ________ 16.
= ________
17.
= ________ 18.
= ________
19.
= ________ 20.
= ________
and
= ___,
= ___
x = ___,
are supplementary, complete the following.
= 5y – 3,
= 2y + 1
26.
Geometry
O
50°
H
G
F
(6x+2)°
are complementary, complete the following.
= 3x,
=x–6
24.
y = ___,
E
30°
(3x+71)°
85°
If
25.
A
D
22.
(2x–3)°
x = ___,
C
B
15.
and
N
= _______
In the diagram,
= 90°,
= 30°, and
Calculate the measures of the following angles.
13.
= ________ 14.
= ________
If
23.
I
D
= 32°, then
Calculate the value of x
21.
E
R
= ___,
= ___
- 38 -
y = ___,
= x + 10,
= ___,
= y – 9,
= ___,
= 2x – 7
= ___
= 4y + 14
= ___
Unit 3 Objective 0
Perimeter, Area, and Volume Review
NOTE: Diagrams are not drawn to scale.
Find the perimeter and area of the figures in problems 1-4.
1.
2.
m
5.
78 in.
5 in.
12 in.
A triangle has a base of 33 yards a height
of 56 yards. Sketch the triangle and find
it’s area.
34 in.
30in.
72 in.
13 in.
6.9 cm
13.5 cm
4.
3.
16 in.
In problems 6-8 use the information about the figure to find the indicated measure.
6.
Perimeter = 84 ft.
Find the length, L
Area = 432 m2
Find the width, w
7.
13 ft
w
24 m
L
8.
Area of shaded triangle = 189 cm2
Find the height, h
h
21 cm
9.
15 cm
The area of a rectangle is 551 square inches, and its width is 19 inches. Find
the length.
Find the volume of the figures below.
10.
11.
5 cm
4 in.
4 in.
4 in.
3 cm
(cont)
Geometry
- 39 -
2 cm
Unit 3 Objective 0 (cont)
12.
A game board is made up of 9 squares put into 3 rows and 3 columns as shown.
Each of the 9 squares has sides that measure 5 cm. Find the perimeter of the
game board.
5 cm
5 cm
The four sides of the figures below will be folded up and taped to make an open
box. What will be the volume of each box?
13.
14.
15. When the box below is closed it has a length of 7 inches, a width of 4 inches and a
height of 5 inches. What is the volume of the box.
Geometry
- 40 -
Geometry Unit 3
1.
Identify pairs of lines and angles. (Section 3.1)
2.
Use parallel lines and transversals. (Section 3.2)
3.
Prove lines are parallel. (Section 3.3)
4.
Two column proofs using parallel line theorems. (Section 3.2 and 3.3)
5.
Find and use slopes of lines. (Section 3.4)
6.
Prove theorems about perpendicular lines. (Section 3.6)
Review
Geometry
- 41 -
Unit 3 Worksheet 3
NOTE: Diagrams are not drawn to scale.
Use the information given to name the lines that must be parallel. If there are no such segments,
write ‘none’
1. m 1 = m 2
2. m 3 = m 4
3. m 5 + m 6 = 180°
a
b
a
b
c
2
1
4.
m
8
4
5.
b
m
9 = m
a
7
d
10
9
6.
m
13 = m
a
14
m
1 = m
k
b
c
10.
7 = m
k
8
11.
m
n
Geometry
9 + m
k
n
8
h
f
2
9. m
4 = m
c
m
3
10 = 180°
h
12. m
f
m
9
10
- 42 -
h
4
6
f
5
6
1 =m
2=m
3
k
h
f
1
2
n
p
p
5= m
k
p
m
7
3
p
m
p
b
d
n
f
h
12
11
n
d
m
2= m
1
m
13
11 = m
d
8.
14
d
12
c
d
7.
m
a
10
c
5
6
b
c
8
b
c
3
d
7 = m
a
a
3
Unit 3 Worksheet 4
c
NOTE: Diagrams are not drawn to scale.
1.
Given:
,
1
r
2
s
Prove:
Statements
1.
1.
2.
is a right angle
3.
2.
= 90°
3.
4.
5.
4.
90° =
6.
5.
is a right angle
6.
7.
2.
Reasons
Given
7.
Given:
a
,
Prove:
b
Statements
3.
1
2
3
Reasons
1.
1.
2.
2.
3.
3.
t
2 1
3 4
Given:
Prove:
is supplementary to
Statements
Reasons
1.
1.
2.
2.
3.
4.
5.
= 180°
= 180°
is supplementary to
3.
4.
5.
(cont)
Geometry
- 43 -
c
6 5
7 8
k
m
Unit 3 Worksheet 4 (cont)
4.
Given:
c
d
3
,
a
2
Prove:
Statements
1.
1.
2.
2.
3.
3.
4.
4.
b
1
Reasons
E
K
5.
Given:
,
2
1
Prove:
L
Statements
4
M 3
T
Reasons
1.
1.
2.
2.
3.
3.
4.
4.
F
6.
Given:
= 90°
= 90°
Prove:
1
W
Statements
Reasons
1.
2.
1.
=
2.
3.
3.
4.
4.
5.
5.
(cont)
Geometry
- 44 -
G
2
M
3
4
L
Unit 3 Worksheet 4 (cont)
D
C
7.
3
Given:
and
are complementary
2
A
Prove:
1.
Statements
and
are complementary
1.
2.
= 90°
2.
3.
3.
4.
= 90°
4.
5.
=
5.
6.
= 90°
6.
7.
8.
=
7.
8.
9.
8.
Given
E
,
V
2
Prove:
1
D
Statements
Reasons
1.
1.
2.
2.
3.
B
Reasons
Given
9.
Given:
1
and
are supplements
4.
3.
= 180°
5.
6.
4.
5.
= 90°
6.
7.
+ 90° = 180°
7.
8.
= 90°
8.
9.
is a right angle
9.
10.
Geometry
10.
- 45 -
Given
A
Unit 3 Worksheet 5
Formulas from Algebra and Geometry can be used to prove a polygon is a particular shape. Use the
distance formula and the formula for the slope of a line for the following coordinate proofs.
DISTANCE =
SLOPE =
=
PARALLEL LINES
have the same slope.
PERPENDICULAR LINES
Example:
=
A (2, -1)
have slopes that are opposite reciprocals.
B (-2, 4)
AB =
Slope of AB =
=
=
=
Slope of any line parallel to AB is
Slope of any line perpendicular to AB is
A quadrilateral is a polygon with 4 sides.
A parallelogram is a quadrilateral with opposite sides parallel.
A rectangle is a parallelogram with four right angles. (Sides are perpendicular)
A rhombus is a parallelogram with four congruent sides.
A square is a parallelogram with four congruent sides and four right angles.
A trapezoid is a quadrilateral with exactly one pair of parallel sides.
For each problem,
•
Plot the points and connect in order to form a quadrilateral.
•
Find the length (distance) of each of the four sides, show your work and organize it neatly!
•
Find the slope of each of the four sides, show your work neatly!!
•
State the facts (which sides are congruent, which sides are parallel, which sides are perpendicular) to
prove what type of polygon the shape is, use as many of the above names as fit.
1.
A (2, 3), B (5, 1), C (2, –1), D (–1, 1)
2.
N(–4, 1), E (–1, 3 ), A( 3, –3 ), T( 0, –5 )
(cont)
Unit 3 Worksheet 5 (cont)
Geometry
- 46 -
3.a.
b.
c.
Find the slopes of SC and CB
Calculate the product of their slopes
Is
4.
? Justify your answer.
S
R
Given: Quad. LAMB as shown.
a. Find the slopes of LA and MB
b. Is
Justify your answer.
B
C
M
A
L
A
B
5.
Determine if
Justify your answer
6.
Given: Quad. CAGE as sketched.
Justify all answers
a. Determine if
b.
c. Determine the length of
d. Determine the length of
D
A
Determine if
C
A
B
C
E
Unit 3 Worksheet 6
Geometry
- 47 -
G
1.
Given:
The slope of line m is
.
line m
line p
Which statement below must be true?
2.
3.
A.
The slope of line p is
B.
The slope of line p is
C.
The slope of line p is
D.
The slope of line p is
Which statement would prove that
A.
(the length of KF) = (the length of TV)
B.
(the slope of KF) = (the slope of TV)
C.
(the slope of KF) =
D.
(the slope of KF) = – (the slope of TV)
All the statements below, except one, will prove that
statement will NOT prove they are perpendicular?
A.
(slope
B.
slope
C.
D.
4.
slope
slope
) • (slope
=
)= –1
–
=
=
Given:
V
F
T
K
x
is perpendicular to
y
. Which
G
L
T
–
and
the slope of
Which one statement below is true?
A.
The slope of
is
B.
The slope of
is
C.
The slope of
is
D.
The slope of
is –
(cont)
Unit 3 Worksheet 6 (cont)
Geometry
y
- 48 -
x
5.
Given:
and
are two distinct lines.
The slope of
,
the slope of
Which statement below must be true?
A.
B.
6.
C.
and
are skew lines
D.
is more steep than
(slope of line r ) • (slope of line v ) = – 1
Given:
and
slope of line r =
Which statement below must be true?
7.
8.
A.
(slope of line v ) =
B.
(slope of line v ) =
C.
(slope of line v ) =
D.
(slope of line v ) =
y
k
Which statement must be true?
A.
(slope k) + (slope t) = 1
B.
(slope k) + (slope t) = –1
C.
(slope k) • (slope t) = –1
D.
(slope k) • (slope t) = 1
x
t
y
Which statement below is true?
A.
(slope f) is positive
B.
(slope f) is the same as the (slope g)
C.
(slope f) is greater than (slope g)
D.
(slope f) • (slope g) = – 1
f
x
g
9.
Given: slope of
=
,
slope of
= –
Which statement must be true?
A.
B.
C.
D.
Unit 3 Review
Geometry
- 49 -
,
slope of
=
Note: Diagrams are not drawn to scale.
In problems 1 – 6, identify the pairs of angles below as:
A. Corresponding
B. Alternate Interior
D. Consecutive Interior (Same Side Interior)
1.
3 and
6
2.
2 and
7
3.
4 and
8
4.
5 and
8
5.
3 and
5
6.
1 and
8
C. Alternate Exterior
E. Vertical
1 2
3 4
5 6
7 8
In problems 7 - 14 solve for the missing variables. Show all work.
7.
8.
9.
24°
5(x – 9)°
65°
(3x + 11)°
(2x – 18)°
(3x + 15)°
10.
11.
2x°
12.
5x°
(4y – 4)°
(4y – 10)°
4x°
(5x + 6y)°
y°
40°
z°
(3x + 6y)°
13.
14.
120°
z°
2x° y° 50°
32°
y°
x°
130°
(cont)
Geometry
- 50 -