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Geometry
Midterm Review (B/C-Level) ANSWER KEY
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Chapter 1: Foundations of Geometry
1) Name the Plane:
Plane FAR
R
2) Name 3 collinear points:
R, I, A
I
A
3) Name 3 non-collinear points:
F, R, A
F
4) Name all the rays in the plane:

, 
, 
, 
, 
AF
AI
AR
RI
RA
5) If E is the midpoint of DF , DE = 2x + 4 and EF = 3x -1. Find DE, EF, and DF.
DE =____14u_____________
EF=_____14u_____________
DF=____28u______________
6) Find the length of KL.
M
x
L
2.5x
K
_____________5x – 3 _______________
KL=_______5u_____________
2
7) T is in the interior of <PQR. Find each of the following. (Hint: Draw a diagram)
a. m<PQT if m<PQR= 35º and m<RQT =10 º.
_______25°______________
b. m<PQR if m<PQR= (2x + 35)º, m<RQT = (x – 5 )º and
m<PQT = (6x +10)º
________47°______________
c. m<PQR if QT bisects <PQR, m<RQT = (3x+8)º and m<PQT = (9x – 4 )º
________28°_____________
8) <DEF and <FEG are complementary. m<DEF = (5y + 1) º, and
m<FEG = (3y – 7 )º. Find the measure of both angles.
m<DEF=___61°___________
m<FEG=___29°___________
9) <DEF and FEG are supplementary. m<DEF = (3z+12) º and m<FEG = (7z – 32 )º.
Find the measures of both angles.
m<DEF=____72°________
m<FEG=____108°__________
10) Use your knowledge of vertical angles to solve for x.
x=5
11) What is the angle measure of <1 & < 2?
165°
(2x + 5)°
1
2
(4x – 5)°
3
12) Find a counterexample to show the conjecture is false. “Any number divisible by
two is also divisible by 4.”
A)
8
B)
16
C) 18
D) 20
13) If two lines intersect, they intersect in exactly ___1 point_____________________.
14) If two planes intersect, they intersect in exactly ___1 line____________________.
15) Find the circumference and area of the circle with radius of 25. Use 3.14 for pi.
Round to nearest hundredth if necessary.
C = __157 m ______
A = _1,962.5 m2___
16) Find the area of the polygon.
12 ft
5ft
9ft
15ft
A = _165 sq. feet ___
17) If <A and <B are supplementary angles and m< A = ½ m<B, find m<A and m<B.
m<A = ___60°_____
m< B = _120°_______
18) Bisect the <ABC (Answers will vary)
A
B
C
4
Chapter 2: Geometry Reasoning
19) Write the converse, inverse, and contra positive of the conditional statement “If
Stephanie’s birthday is January 1st, then she was born on New Year’s Day.” Find
the truth value of each.
Converse: If Stephanie was born on New Year’s Day then Stephanie’s birthday is
January 1st. (True)
Inverse : If Stephanie’s birthday is not January 1st then she was not born on New
Year’s Day. (True)
Contra positive: If Stephanie was not born on New Year’s Day then Stephanie’s
birthday is not January 1st. (True)
20) For the conditional “If an angle is straight, then its measure is 180 degrees,” write
the converse and the bi-conditional.
Converse: If the angle measures 180 degrees then it is a straight angle.
Bi-conditional: An angle is straight if and only if its measure is 180 degrees.
21) Determine if the bi-conditional “x2 = 100 if and only if x = 10” is true. If false, give
a counterexample.
False. Counterexample: -10
22) Test the statement to see if it is reversible. If so, write as a true bi-conditional
statement. If not, write not reversible. “If lines intersect they intersect in exactly
one point” is true. If false, give a counterexample.
True
Lines intersect if and only if they intersect in exactly one point.
23) Write the conditional, converse, inverse and contra positive of the statement: All
rectangles have four right angles.
Conditional: If the figure is a rectangle then it has four right angles.
Converse: If the figure has four right angles then it is a rectangle.
Inverse: If the figure is not a rectangle then it does not have four right angles.
Contra positive: If the figure does not have four right angles then it is not a rectangle.
5
24) Write the definition “A scalene triangle is a triangle with three different side
lengths” as a bi-conditional.
A triangle is scalene if and only if it has three different side lengths.
25) Change the following statement to a conditional statement: All even numbers are
divisible by 2.
If the number is even then the number is divisible by 2.
26) Identify the hypothesis and conclusion of the conditional. If a triangle has one
angle greater than 90 degrees then it is an obtuse triangle.
Hypothesis: A triangle has one angle greater than 90°
Conclusion: It is an obtuse triangle
27) Write the converse of the statement: “If it is Memorial Day, I do not have to go to
school.”
If I do not have to go to school then it is Memorial Day.
28) Are the following statements true or false? If false, provide a counterexample.
a. If it is Monday, then I have to go to school.
F, Snow Day
b. If you have two right angles, then the angles are congruent.
True
c. If a number is divisible by 3, then it is also divisible by 9.
False, 3 or 6
d. If you eat a piece of fruit, then it must have seeds.
True
Chapter 3: Parallel and Perpendicular lines
29) Name all the segments that are parallel to BC
FG, AD, EH
E
F
C
D
30) Name all segments that are perpendicular to BC
BG, BA
31)
H
Name a pair of skew lines. BC & HG
A
32) Name a pair of parallel planes. Plane DCEB and Plane EFGH
G
B
6
k
1
2
m
3 4
5 6
7 8
mn
n
Use the diagram above for questions # 33 - 44
33) Name all pairs of vertical angles.
1 & 4;  2 & 3;  5 &  8;  6 & 7
34) Name all pairs of same side interior angles
3 &  5; 4 &  6
35) Name all pairs of corresponding angles.
3 & 7; 1 &  5;  2 &  6; 4 & 8
36) Name all pairs of alternate interior angles.
 3 &  6;  4 &  5
37) Name all pairs of alternate exterior angles.
 1 & 8;  2 &  7
38) Name all pairs of same side exterior angles.
 1& 7; 2 & 8
39) Name all angles that are supplementary to  1.
2,  3, 6; <7
40) Line k is a transversal line of m and n. Name a pair of angles whose equality would
guarantee that line m is parallel to line n.
Angle pair of angles except vertical angles would prove you have parallel lines.
Then find the angle measures. Use the diagram above.
41) m4 = (8x – 34 )°; m5 = (5x + 2)°
m 4 = 62° ________
m5 = 62° ________
42) m1 = (23x + 11)°; m7 = (14x + 21)°
m1 = 103° _______
m7 = 77° ________
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43) m2 = (7x – 14)°; m6 = (4x + 19)°
m 2 = 63° ________
m6 = 63° ________
44) mv1 = (6x + 24)°; m4 = (17x – 9) °
m 1 = 42° _______
m 4 = 42° _______
s
r
7
1
8
6
4 5
2 3
Use the diagram above for the following questions. Use the theorems and given
information to show that r s .
45) 1  5
Converse of Alternate Exterior Angle Thm
46) m3  m4  180
Converse of Same-Side Interior Angle Thm
47) 3  7
Converse of Alternate Interior Angle Thm.
8
48) m4  (13x  4); m8  (9 x  16); x  5
m4 = 61° ______________
m 8 = 61° ______________
Converse of Alternate Interior Angle Thm
49) m8  (17 x  37); m7  (9 x  13); x  6
m8 = 139° _____________
m7 = 41° ______________
Converse of Same-Side Interior Angle Thm
50) m2  (25 x  7); m6  (24 x  12); x  5
m2 = 132° _____________
m6 = 132° _____________
Converse of Alternate Exterior Angle Thm
51) Given: p q
Prove: m1  m3  180
1
Statement
1
Given
pq
p
2 m< 2 + m< 3 =
180°
3 1  2
Linear Pair Theorem
4 m1 = m2
Definition of Congruent
Angles
Substitution
5
m1  m3  180
2
3
Reason
q
Corresponding  Postulate
52) Given: l m, 1  3
1
Prove: r p
l
Reason
Statement
2
p
3
r
m
1
l m,
Given
2
1  3
Given
3
1  2
Corresponding Angle Postulate
4
2  3
Transitive Property of Congruence
5
r p
Converse of Alternate Exterior Angle
Theorem
9
Chapter 4: Triangle Congruencies
53) Label each diagram appropriately and identify which triangle congruence theorem
satisfies the diagram. Choose from SSS, SAS, ASA, AAS, HL or not possible. If it
is not possible explain why. Show all work; put a box around your answer.
Triangles are not drawn to scale.
a. Prove ABD  ACD
A
B
SSS
C
D
b. Prove ADC  ABC
D
C
A
B
ASA
c. Prove ABC  DEC
A
A
D
A
B-Level: HL
B
(C-Level: Not possible) A
d. Prove ABC  DEC
A
B
C
ASA
E
E
A
C
A
D
B
C
54) Given: AC bisects BD
BD bisects AC
E
A
Prove: ΔAEB  ΔCED
D
Statement
1
AC bisects BD
2
BD bisects AC
10
Reason
Given
Given
3
DE  BE
Definition of Segment Bisector
4
AE  CE
Definition of Segment Bisector
5
AEB  DEC
Vertical Angle Theorem
6
ΔAEB  ΔCED
SAS
B
55) Given: AB  BC
BD  AC
A
D
Prove: ABD  CBD
Statement
Reason
1
AB  BC
Given
2
BD  AC
Given
3
BDA and BDC are right angles
Definition of Perpendicular Lines
4
BDA  BDC
Right Angle Theorem
5
BD  BD
Reflexive Property of Congruence
6
ABD  CBD
HL
C
11
M
56) (B-Level) Given:
MJ  NJ
MJK  NJK
Prove: MK  NK
K
J
N
Reason
N
Statement
1
MJ  NJ
Given
2
MJK  NJK
Given
3
JK  JK
Reflexive Property of Congruence
4
MJK  NJK
SAS
5
MK  NK
CPCTC
M
(C-Level) Given:
MJ  NJ
MJK  NJK
Prove: JMK  JNK
K
J
N
Statement
Reason
N
1
MJ  NJ
Given
2
MJK  NJK
Given
3
JK  JK
Reflexive Property of Congruence
4
MJK  NJK
SAS
12
57) Find the measure of each angle. 31°,71°,78°
(2x+19)º
(x+5)°
3x°
58) Find the measure of x. x = 50°
(2x + 3)°
(4x – 7 )°
Chapter 5: Properties and Attributes of Triangles
59) Find the measure of <R and <P.
P
R
(2x-10)º
m<R:___14°________
(4x -34)º
Q
m<P:_152°_________
60) Find the value of x. x = 60°
(x)º
13
Decide which of the given side lengths will form a triangle when constructed.
Support your answer using the Triangle Inequality Theorem.
61) 6 ft, 8 ft, 10 ft
Yes it will form a triangle
62) 4 m, 5 m, 9 m
No it will not form a triangle
63) 11 cm, 14 cm, 17 cm
Yes it will form a triangle
64) Solve for the missing side. Round to nearest tenth if necessary.
8 ft
3 ft
8.5 ft
.
65) Solve for the missing side. Round to the nearest tenth when necessary.
20 ft
15 ft
13.2 ft
14
66) Sir Shrek is off to rescue Princess Fiona in the highest tower of the castle. He
shoots an arrow with a 75 foot rope attached to it, to the top of the tower. Shrek is
stand 20 feet away from the tower when he shoots the arrow. How tall is the tower
he has to climb in order to rescue Princess Fiona? Round to the nearest tenth.
72.2 ft
67) Find the midpoint of the following ordered pairs.
a) A(1,2) & B(6, 8) (3.5, 5)
Midpoint Formula
 x1  x 2 y1  y 2 
,


2 
 2
b) C(0,-6) & D(4, 0) (2, -3)
c) E(-4,-12) & F(6, -8) (1, -10)
68) If ABC  DEF , state all corresponding segments and angles that are congruent.
 A  D;  B  E;  C  F
AB  DE ; BC  EF ; AC  DF
69) Solve for x. Round to the nearest tenths.
9
12
x = 7.93 units
60°
60°
x
70) Find the length of line segment CD , if AB = 26; AE = 10 and m<C = 45°. Round to
the nearest tenth.
B
C
CD = 33.94 units
A
E
F
D