Midterm Exam
PMI Geometry
Name ________________________
Multiple Choice– Choose the correct answer for each question. No partial credit will be given.
1. If points B, D, and X are collinear, with D between X and B. Given: DX = 4x + 6, BX = 6x + 18, and
DB = x + 15, solve for x.
a. x = -1
b. x = 18
c. x = -3
d. x = 3
2. ⃗⃗⃗⃗⃗
XB lies on the interior of ∠𝑊𝑋𝑍, if 𝑚∠𝑊𝑋𝐵 = 30° and 𝑚∠𝑍𝑋𝑊 = 40°, what is the 𝑚∠𝐵𝑋𝑍?
a. 70°
b. 35°
c. 10°
d. 5°
3. C is between A and B. If AC = 4x, AB = 3x + 12, and CB = 10, then AB =
a. 2
b. 18
c. 12
d. -2
⃗⃗⃗⃗⃗ 𝑏𝑖𝑠𝑒𝑐𝑡𝑠 ∠𝐴𝐵𝐶. Find x if 𝑚∠𝐴𝐵𝑋 = (2𝑥 − 3)°
4. ∠𝐴𝐵𝐶 is a right angle. 𝐵𝑋
a. 87
b. 45
c. 24
d. 21
5. The figure shows the o, p, and q intersecting to form the angles numbered 1, 2, 3, 4, 5, and 6. All of
the lines are in the same plane. Based on the figure, which of the individual statements would
provide enough information to concluded that line o is perpendicular to line p? Circle all that apply.
a.
b.
c.
d.
𝑚∠3 + 𝑚∠4 = 90°
𝑚∠3 = 90°
𝑚∠5 + 𝑚∠6 = 90°
𝑚∠4 + 𝑚∠5 = 90°
e. 𝑚∠3 = 𝑚∠6
f. 𝑚∠4 = 90°
p
o
q
1 23
6 54
6. Tell whether the lengths 8, 14, and 10 form the sides of an acute, right, obtuse, or a not a triangle.
a. acute
b. right
c. obtuse
d. not a triangle
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E
7. ∆𝐵𝐶𝐷~∆𝐸𝐴𝐹. Find the value of x.
a. 8
b. 9
c. 16
d. 18
x
4
B
12
A
6
F
6
C
9
D
8. The legs of a right triangle are 4 and 7, what is the approximate length of the hypotenuse?
a. 5.7
b. 8.1
c. 9.4
d. 10.0
9. Find the value of x.
B
(3x + 20)°
A
a.
b.
c.
d.
(5x + 10)°
C
D
40°
12
20
25
50
10. Which of the following sets of lengths could not be the sides of a triangle?
a. 12, 8 , 7
b. √7, √5, √2
c. 10, 10, 20
d. 3,5,6
11. What is 𝑚∠𝐴?
a. 8
b. 12
c. 24
d. 40
A
(2x)°
80°
(3x + 20)°
60°
12. What is the value of x?
a. 165
b. 155
c. 140
d. 120
x°
140°
A
13. Given the triangle at right. It is:
a. Isosceles and right
b. Scalene and obtuse
c. Equiangular and equilateral
d. Scalene and acute
52°
65°
C
B
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For Questions 14 and 15, use the diagram at right
14. Given 𝑘 ∥ 𝑙, 𝑚∠7 = (4𝑥 − 20)°, 𝑎𝑛𝑑 𝑚∠1 = (6𝑥 − 40)°, find 𝑚∠6
a. 10°
b. 20°
c. 24°
d. 104°
15. Given 𝑚 ∦ 𝑛, 𝑚∠12 = (3𝑥 − 20)°, 𝑎𝑛𝑑 𝑚∠8 = (2𝑥 + 20)°, 𝑡ℎ𝑒𝑛 𝑥 ≠
a. 44
b. 40
c. 39
d. 36
16. The figure shows line p, points G, J and I are on line j, and point H is not on line j. Also shown is line
HJ.
H
p
G
J
I
Part A:
K
p
G
H
J
I
Consider the partial construction of a line parallel to p through point H. What would be the final step in
the construction?
a. Draw a line through points K and H
b. Draw a line through points I and K
c. Draw a line through points G and K
d. Draw a line through points G and H
Part B:
Once the construction is complete, which of the following reasons listed contribute to providing the validity
of the construction?
a. If two parallel lines are cut by a transversal, then the corresponding angles are congruent.
b. If two parallel lines are cut by a transversal, then the same-side interior angles are
supplementary.
c. If two parallel lines are cut by a transversal, then the alternate exterior angles are congruent.
d. If two parallel lines are cut by a transversal, then the alternate interior angles are congruent.
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17. 𝑘 ∥ 𝑚 then the reason ∠1 = ∠2
a. they are alternate interior angles
b. they are alternate exterior angles
c. they are same side angels
d. they are corresponding angles
J
̅̅̅ ?
18. What is the length of 𝐽𝐾
a. 8.00
b. 10.24
c. 16.50
d. 16.64
13
K
52°
L
Short Constructed Response – Write the correct answer for each question. No partial credit will be
given.
19. Name the segments that are skew to
EF
20. You want to find the height of a tall building that is casting a shadow 40 feet long. At the same time,
you are casting a shadow that is 2 feet long. If you are 5 feet 6 inches tall, then how tall is the
building?
21. Find the measurements of both acute angles in the triangle below. Round your answers to the
nearest hundredth.
N
19
O
37
M
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Extended Constructed Response - Solve the problem, showing all work. Partial credit may be given.
22. Complete the proof by filling in the missing reasons with the “reasons bank” to the right. Some
reasons may be used more than once and some may not be used at all.
Q
T
Reasons Bank
a) If 2 parallel lines are cut by a transversal,
then the alternate interior angles are
congruent.
b) If 2 parallel lines are cut by a transversal,
then the corresponding angles are
congruent.
c) Given
d) Vertical angles are congruent
e) AA~
f) SAS~
g) SSS~
R
P
S
̅̅̅
Given: ̅̅̅̅
𝑃𝑄 || ̅𝑆𝑇
Prove: ∆𝑃𝑄𝑅~∆𝑇𝑆𝑅
Statements
̅̅̅
1. ̅̅̅̅
𝑃𝑄 || ̅𝑆𝑇
2. ∠𝑄 ≅ ∠𝑆
3. ∠𝑃𝑅𝑄 ≅ ∠𝑇𝑅𝑆
4. ∆𝑃𝑄𝑅~∆𝑇𝑆𝑅
Reasons
1.
2.
3.
4.
23. You are on the deck of a cruise ship. Your eyes are 500 m above the sea level. You look down
towards the ocean and observe two seals. The angle of depression from the ship is 32° & 33°,
respectively.
a) How far is the first seal from the ship?
b) How far is the second seal from the ship?
c) What is the distance between the 2 seals?
d) If the cruise ship stops at one of its destinations & the seals continue to travel toward the ship
at a rate of 5 meters per second, how long will it take them to get to the ship?
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24. The points J(-1, 7), K(2, 3), and L(-1, 3) form a triangle.
a. Graph the triangle in the coordinate plane to the right.
b. Based on the sides, what type of triangle is formed?
c.
Base on the angles what type of triangle is formed?
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Part 2 – Midterm Exam
Name:______________________________________
1. Write a proof
Given: 𝑘 || ℎ
Prove: 𝑚∠3 + 𝑚∠7 = 180°
j
1 2
3 4
5 6
8 7
k
h
2. The angle of elevation from the top of a small building to the top of a tall building is 50°, and
the angle of depression from the top of the small building to the base of the tall building is
23°. If the smaller building is 20 m high, how tall is the tall building?
3. In right triangle KLM, 𝑚∠𝐾 ≠ 𝑚∠𝑀. Let 𝑠𝑖𝑛𝐾 = 𝑥 𝑎𝑛𝑑 𝑐𝑜𝑠𝐾 = 𝑦. What is 𝑐𝑜𝑠𝑀 − 𝑠𝑖𝑛𝑀?
Explain your answer.
K
L
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M
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4. Answer ONE of the following:
a. Explain why angle- angle is enough to establish similarity between two triangles.
OR
b. Explain why congruence is a special case of similarity.
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Answers to Geometry Midterm
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
12.
13.
14.
15.
16.
17.
18.
19.
20.
21.
22.
D
C
B
C
C&F
C
A
B
C
C
C
B
D
D
D
Part A: A
Part B: C
D
B
segments AB & CD
110 feet
𝑚∠𝑀 = 59.10° & 𝑚∠𝑂 = 30.9°
Statements
̅̅̅
1. ̅̅̅̅
𝑃𝑄 || ̅𝑆𝑇
2. ∠𝑄 ≅ ∠𝑆
Reasons
1. C. Given
2. A. If 2 parallel lines are cut by a transversal, then the
alternate interior angles are congruent.
3. D. Vertical Angles are congruent
4. E. AA~
3. ∠𝑃𝑅𝑄 ≅ ∠𝑇𝑅𝑆
4. ∆𝑃𝑄𝑅~∆𝑇𝑆𝑅
23. part a) 769.93 m
part b) 800.17 m
part c) 30.24 m
part d) 160.03 seconds
= 2.67 min. & 153.99 seconds = 2.57 min.
8
J
6
24. a.
4
L
–10
2
–5
K
5
10
–2
–4
–6
b. scalene
c. right triangle
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PART 2
1. Many ways to answer this question. Sample proof is given below:
Statements
1. 𝑘 || ℎ
2. ∠4 ≅ ∠7
Reasons
1. Given
2. If 2 parallel lines are cut by a transversal, then the
corresponding angles are congruent.
3. Definition of congruent angles
4. Linear pair angles are supplementary or sum to 180
5. Substitution Property of Equality
3. 𝑚∠4 = 𝑚∠7
4. 𝑚∠3 + 𝑚∠4 = 180°
5. 𝑚∠3 + 𝑚∠7 = 180°
2. See Diagram below
x = 47.11 m
y = 56.15 m
h = 76.15 m
y
20
50° x
23°
h = y + 20
20
3. Since 𝑠𝑖𝑛𝐾 = 𝑥 𝑎𝑛𝑑 𝑐𝑜𝑠𝐾 = 𝑦, KL = y, LM = x & KM = 1 (sine = (opp)/(hyp), so hyp = 1). If you
now take the sine and cosine of angle M and you get the following answers: 𝑠𝑖𝑛𝑀 =
𝑦 𝑎𝑛𝑑 𝑐𝑜𝑠𝑀 = 𝑥. Therefore, 𝑐𝑜𝑠𝑀 − 𝑠𝑖𝑛𝑀 = 𝑥 − 𝑦.
4.
a. once you know that 2 pairs of angles are equal, the 3rd pair has to be equal, because all 3
angles of each triangle add up to 180 degrees.
b. congruence is similarity with a scale factor of 1, which makes the sides similar AND equal.
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