Geometry Essentials (2015-2016)
Midterm Review
Name: _________________________________
Chapter 1
For numbers 1 – 4, use the diagram below.
1. Classify ∠𝐴𝑂𝐷 as acute, obtuse, right or straight.
2. ∠𝐶𝑂𝐷 is a linear pair with what other angle?
3. Name an angle adjacent to ∠𝐵𝑂𝐸.
4. Name an angle vertical to ∠𝐵𝑂𝐴.
5. Find the value of x using the diagram below.
6. Two angles are complementary. The first angle measures (𝑥 + 15)° and the second angle measures (14𝑥)° . Find the
measure of the second angle.
Chapter 2
State the definition, property, postulate, or theorem that justifies each statement.
7. If X is the midpoint of ̅̅̅̅
𝐶𝐷 , then ̅̅̅̅
𝐶𝑋 ≅ ̅̅̅̅
𝑋𝐷 .
8. If ∠K ≅ ∠P and ∠P ≅ ∠T, then ∠K ≅ ∠T.
9. If m∠W + m∠H = 90° and m∠H = 20° , then m∠W + 20 = 90.
10. If ∠A and ∠B are complementary, then mA + mB = 90.
11. If ̅̅̅̅
𝐴𝑇 ≅ ̅̅̅̅
𝐷𝑅 then AT = DR.
12. AB + BC = AC
Chapter 3
̅̅̅.
13. Find the slope of a line that is perpendicular to ̅𝑆𝑇
14. Write an equation in slope-intercept form for the line that passes through the point (−1, 4) and is parallel to the linear
graph of 𝑦 = −2𝑥 + 5.
For numbers 15 – 17, use the diagram to the right.
15. Name the relationship between angles 8 and 14 and tell whether they
are congruent or supplementary.
16. Name the relationship between angles 5 and 13 and tell whether they
are congruent or supplementary.
17. Name the relationship between angles 7 and 14 and tell whether they
are congruent or supplementary.
18. Find the value of x.
19. Find the value of x.
Chapter 4
20. Triangles ABC and PQR are congruent.
Which statement about the triangles is true?
a) A  R
b) C R
c) AB  RQ
d) CB  PQ
21. Which figure contains two congruent triangles?
a)
b)
c)
d)
22. Triangle ABC is congruent to triangle XYZ, as shown below.
Which of the following statements must be true?
a) mX = 45°
b) mZ = 45°
c) YZ  3 cm
d) XY  3 cm
23. If ∆EFG  ∆HIJ, which of the following statements must be true?
a) ∆EFG  ∆JIH
c) ∆FEG  ∆IHJ
b) ∆EFG  ∆IJH
d) ∆FGE  ∆HIJ
24. Consider these statements.
*No equilateral triangles are obtuse.
*Triangle KJF is obtuse.
Which conclusion can be made using both statements?
a) Triangle KJF is scalene.
c) Triangle KJF is a right triangle.
b) Triangle KJF is not isosoceles.
d) Triangle KJF is not equiangular.
25. Jeremy is proving the following theorem:
If two angles are congruent to two angles of a second triangle, then the third angles of the triangles are congruent.
Jeremy’s proof is shown below.
Given: Q  T
P  S
Prove: R  V
Statements
1. Q  T
P  S
2. mQ = mT
mP = mS
3. mP + mQ + mR = 180°
mS + mT + mV = 180°
4. mP + mQ + mR = mS + mT + mV
5. ?
6. mR = mV
7. R  V
Reasons
1. Given
2. Definition of Congruency
3. The interior angles of a triangle add to 180°
4. Transitive Property of Equality
5. Substitution Property
6. Subtraction Property of Equality
7. Definition of Congruency
Which statement BEST represents Step 5 of Jeremy’s proof?
a) 180° = 180°
c) mP + mQ = mT + mS
b) 180° – mR = 180° – mV
d) mP + mQ + mR = mP + mQ + mV
26. Manuel is trying to prove the following theorem. If two sides of a triangle are congruent, then the angles opposite
these sides are congruent. First Manuel draws isosceles ∆PQR, and then he adds an auxiliary line that bisects PQR.
Q
An incomplete version of Manuel’s proof is shown below.
Statements
1. PQ = RQ
2. mPQS = mRQS
3. QS = QS
4. ?
5. mQPS = mQRS
Reasons
1. Given
2. QS bisects PQR
3. Reflexive Property
4.
5.
What should be the statement for Step 4 of Manuel’s proof?
a) PQR is a right angle
c) PSQ  RSQ
b) ∆PQS  ∆RQS
d) PS  SR
P
S
R
27. The statements for a proof are shown.
Statements
1. Parallelogram ABCD
1  2
2. B  D
3. AB  DC
4. ∆ABX  ∆CDY
5. BX  DY
Reasons
1. Given
2.
3.
4.
5.
What is the reason that the statement in Step 5 is true?
a) angle-side-angle
b) side-angle-side
c) Opposite sides of a parallelogram are congruent
d) Corresponding parts of congruent triangles are congruent (CPCTC)
28. A triangle is shown.
Which triangle is congruent to this triangle?
a)
b)
c)
29. Triangle EFG is congruent to Triangle HIJ.
What is the measure of Angle IJH?
a) 24°
b) 80°
c) 156°
d) 204°
30. Triangles ADC and BCD are shown below.
What additional fact is needed to prove ∆ADC  ∆BCD by the hypotenuse leg theorem?
a) A  B
b) AC  BD
c) AD  BC
d) mA = mB = 45°
d)
31. In the figure below, ∆XYZ  ∆XWZ.
What is the length of XY ?
a) 7
b) 9
c) 10
d) 12
32. In the figure to the right, QR  SR and PR  TR.
Based only on the given information, which theorem could be used to prove ∆PQR  ∆TSR?
a) Side-Side-Side
b) Side-Angle-Side
c) Side-Side-Angle
d) Angle-Side-Angle
33. If ∆OMR  ∆STP, which relationship must be true.
a) RM  TS
c) MRO  PST
b) RO  PS
d) RMO  TPS
34. In the figure shown, M  T, and R is the midpoint of MT.
To prove MN  TP, a student wrote the following 6 statements.
1. M  T
2. R is the midpoint of MT
3. MR = TR
4. MRN  TRP
5. ∆MRN  ∆TRP
6. MN  TP
Which reason should the student give for statement 5?
a) Side-Angle-Side
c) Angle-Side-Angle
b) Side-Angle-Angle
d) Angle-Angle-Angle
35. A proof is shown below.
Given: LM || PQ and MT  TP
Prove: ∆LMT  ∆QPT
STATEMENT
REASON
1. LM || PQ and MT  TP
2. MLT  PQT
3. LTM  QTP
4. LMT  QTP
1. Given
2. Alternate Interior Angles Theorem
3. __________________
4. Angle-Side-Angle Theorem
Which reason is justification for statement 3?
a) Angle Bisector Theorem
c) definition of congruent angles
b) Vertical Angles Theorem
d) definition of congruent triangles
36. In the figure below, PQ  RQ and PS  RS.
Based only on the given information, which theorem could be used to prove ∆PQS  ∆RQS?
a) Side-Side-Side
c) Side-Angle-Side
b) Side-Side-Angle
d) Angle-Side-Angle
37. ∆ABC is shown. Which value of x will prove △ABC is a right triangle?
a)
b)
c)
d)
A
x = 10
x = 30
x = 45
x = 91
(3x)°
(2x + 1)°
B
(x – 1)°
C
38. Which statement describes a triangle that must be scalene?
a) A triangle with all sides congruent.
c) A triangle with 2 sides congruent.
b) A triangle with all angles congruent.
d) A triangle with no sides congruent.
39. Which statement is NOT enough information to prove the two triangles shown are congruent?
a) A  D and B  E
b) AC  DF and A  D
c) AC  DF and AB  DE
d) AC  DF and B  E
40. Which expression represents the missing x-coordinate for S?
0  3b
2
a) 3b
b)
c) 0 – 3b
d) cannot be determined
41. XY is the hypotenuse of right ∆XYZ. Which of the following could be the coordinates of Z?
a) (0, 0)
b) (–2, 2)
c) (–1, 2)
d) (0, 2)