Download Geo Final Review 2014

Name: ________________________ Period: ___________________ Date: __________
Geo Final Review 2014
Multiple Choice
Identify the choice that best completes the statement or answers the question.
____
1. An angle measures 2 degrees more than 3 times its complement. Find the measure of its complement.
a. 23°
c. 22°
b. 272°
d. 68°
____
2. BD bisects ∠ABC , m∠ABD = (7x − 1)°, and m∠DBC = (4x + 8)°. Find m∠ABD.
a. m∠ABD = 22°
c. m∠ABD = 20°
b. m∠ABD = 3°
d. m∠ABD = 40°
____
3. Use the information m∠1 = (3x + 30)°, m∠2 = (5x − 10)°, and x = 20 , and the theorems you have learned to
show that l Ä m.


→
a.
b.
c.
d.
By substitution, m∠1 = 3(20) + 30 = 90° and m∠2 = 5(20) − 10 = 90° .
Since ∠1 and ∠2 are same-side interior angles, m∠1 = m∠2 = 180° .
By the Converse of the Same-Side Interior Angles Theorem, l Ä m.
Since ∠1 and ∠2 are same-side interior angles, m∠1 = 3(20) + 30 = 90° and
m∠2 = 5(20) − 10 = 90° .
By substitution, m∠1 = m∠2 = 90° .
By the Converse of the Alternate Interior Angles Theorem, l Ä m.
By substitution, m∠1 = 3(20) + 30 = 90° and m∠2 = 5(20) − 10 = 90° .
By the Substitution Property of Equality, m∠1 = m∠2 = 90° .
By the Converse of the Alternate Interior Angles Theorem, l Ä m.
By substitution, m∠1 = 3(20) + 30 = 90° and m∠2 = 5(20) − 10 = 90° .
Since ∠1 and ∠2 are alternate interior angles, m∠1 = m∠2 = 180° .
By the Converse of the Same-Side Interior Angles Theorem, l Ä m.
1
Name: ________________________
____
ID: A
4. In a dance performance, four dancers form a diamond with vertices A(2, 0), B(0, 2), C(−2, 0), and D(0, − 2).
Then, they move along the dance floor following the translation vector, 0, 4 . There they pause, and then
move again along the same vector. What are their coordinates after six such translations?
a. A′(26, 0), B ′(24, 2), C ′(22, 0), and D ′(24, − 2)
b. A′(2, 10), B ′(0, 12), C ′(−2, 10), and D ′(0, 8)
c. A′(2, 24), B ′(0, 26), C ′(−2, 24), and D ′(0, 22)
d. A′(26, 24), B ′(24, 6), C ′(22, 24), and D ′(24, 22)
____
5. Two sides of an equilateral triangle measure (2y + 3) units and (y 2 − 5) units. If the perimeter of the triangle is
33 units, what is the value of y?
a. y = 4
c. y = 7
b. y = 11
d. y = 15
2
Name: ________________________
____
ID: A
6. Use the given paragraph proof to write a two-column proof.
Given: ∠BAC is a right angle. ∠1 ≅ ∠3
Prove: ∠2 and ∠3 are complementary.
Paragraph proof:
Since ∠BAC is a right angle, m∠BAC = 90° by the definition of a right angle. By the Angle Addition
Postulate, m∠BAC = m∠1 + m∠2 . By substitution, m∠1 + m∠2 = 90° . Since ∠1 ≅ ∠3, m∠1 = m∠3 by the
definition of congruent angles. Using substitution, m∠3 + m∠2 = 90° . Thus, by the definition of
complementary angles, ∠2 and ∠3 are complementary.
Complete the proof.
Two-column proof:
Statements
1. ∠BAC is a right angle. ∠1 ≅ ∠3
2. m∠BAC = 90°
3. m∠BAC = m∠1 + m∠2
4. m∠1 + m∠2 = 90°
5. m∠1 = m∠3
6. m∠3 + m∠2 = 90°
7. ∠2 and ∠3 are complementary.
Reasons
1. Given
2. Definition of a right angle
3. [1]
4. Substitution
5. [2]
6. Substitution
7. Definition of complementary angles
a.
c.
b.
[1] Angle Addition Postulate
[2] Definition of congruent angles
[1] Angle Addition Postulate
[2] Definition of equality
d.
[1] Substitution
[2] Definition of equality
[1] Substitution
[2] Definition of congruent angles
____
7. Find the measure of the complement of ∠M , where m∠M = 31.1°
a. 58.9°
c. 121.1°
b. 148.9°
d. 31.1°
____
8. One of the acute angles in a right triangle has a measure of 34.6°. What is the measure of the other acute
angle?
a. 90°
c. 34.6°
b. 145.4°
d. 55.4°
3
Name: ________________________
____
ID: A
9. Draw two lines and a transversal such that ∠1 and ∠2 are alternate interior angles, ∠2 and ∠3 are
corresponding angles, and ∠3 and ∠4 are alternate exterior angles. What type of angle pair is ∠1 and ∠4?
a.
∠1 and ∠4 are corresponding angles.
b.
∠1 and ∠4 are supplementary angles.
c.
∠1 and ∠4 are vertical angles.
d.
∠1 and ∠4 are alternate exterior angles.
4
Name: ________________________
ID: A
____ 10. Write a justification for each step.
m∠JKL = 100°
m∠JKL = m∠JKM + m∠MKL
100° = (6x + 8)° + (2x − 4)°
100 = 8x + 4
96 = 8x
12 = x
x = 12
a.
b.
c.
d.
[1]
Substitution Property of Equality
Simplify.
Subtraction Property of Equality
[2]
Symmetric Property of Equality
[1] Angle Addition Postulate
[2] Simplify.
[1] Angle Addition Postulate
[2] Division Property of Equality
[1] Segment Addition Postulate
[2] Multiplication Property of Equality
[1] Transitive Property of Equality
[2] Division Property of Equality
____ 11. The figure shows part of the roof structure of a house. Use SAS to explain why ∆RTS ≅ ∆RTU .
Complete the explanation.
It is given that [1]. Since ∠RTS and ∠RTU are right angles, [2] by the Right Angle Congruence Theorem. By
the Reflexive Property of Congruence, [3]. Therefore, ∆RTS ≅ ∆RTU by SAS.
a. [1] ST ≅ UT
c. [1] ST ≅ UT
[2] ∠RTS ≅ ∠RTU
[2] ∠SRT ≅ ∠URT
[3] RT ≅ RT
[3] ST ≅ UT
b. [1] ST ≅ UT
d. [1] RT ≅ RT
[2] ∠RTS ≅ ∠RTU
[2] ∠SRT ≅ ∠URT
[3] SU ≅ SU
[3] ST ≅ UT
29
Name: ________________________
ID: A
____ 12. Tell whether the figure is a polygon. If it is a polygon, name it by the number of its sides.
a.
b.
polygon, decagon
polygon, hexagon
c.
d.
polygon, dodecagon
not a polygon
____ 13. Use the Converse of the Corresponding Angles Postulate and ∠1 ≅ ∠2 to show that l Ä m.
a.
b.
c.
d.
By the Converse of the Corresponding Angles Postulate, ∠1 ≅ ∠2. From the diagram,
l Ä m.
∠1 ≅ ∠2 is given. From the diagram, ∠1 and ∠2 are alternate interior angles. So by the
Converse of the Alternate Interior Angles Postulate, l Ä m.
∠1 ≅ ∠2 is given. From the diagram, ∠1 and ∠2 are corresponding angles. So by the
Corresponding Angles Postulate, l Ä m.
∠1 ≅ ∠2 is given. From the diagram, ∠1 and ∠2 are corresponding angles. So by the
Converse of the Corresponding Angles Postulate, l Ä m.
____ 14. Find m∠DCB, given ∠A ≅ ∠F , ∠B ≅ ∠E , and m∠CDE = 46°.
a.
b.
m∠DCB = 44°
m∠DCB = 134°
c.
d.
6
m∠DCB = 67°
m∠DCB = 46°
Name: ________________________
ID: A
____ 15. Determine if you can use ASA to prove ∆CBA ≅ ∆CED. Explain.
a.
b.
c.
d.
AC ≅ DC is given. ∠CAB ≅ ∠CDE because both are right angles. By the Vertical
Angles Theorem, ∠ACB ≅ ∠DCE . Therefore, ∆CBA ≅ ∆CED by ASA.
AC ≅ DC is given. ∠CAB ≅ ∠CDE because both are right angles. By the Adjacent
Angles Theorem, ∠ACB ≅ ∠DCE . Therefore, ∆CBA ≅ ∆CED by ASA.
AC ≅ DC is given. ∠CAB ≅ ∠CDE because both are right angles. By the Vertical
Angles Theorem, ∠ACB ≅ ∠DCE . Therefore, ∆CBA ≅ ∆CED by SAS.
AC ≅ DC is given. ∠CAB ≅ ∠CDE because both are right angles. No other congruence
relationships can be determined, so ASA cannot be applied.
____ 16. Translate the triangle with vertices A(3, 4), B(2, − 1), and C(4, 12) along the vector −1, 3 . Find the
coordinates of the new image.
a. A′(2, 7), B ′(1, 2), and C ′(3, 15)
b. A′(4, 7), B ′(3, − 2), and C ′(5, 15)
c. A′(6, 3), B ′(5, − 2), and C ′(7, 11)
d. A′(−3, 12), B ′(−2, − 3), and C ′(−4, 36)
7
Name: ________________________
ID: A
____ 17. Given: ∠MLN ≅ ∠PLO, ∠MNL ≅ ∠POL, MO ≅ NP
Prove: ∆MLP is isosceles.
Complete the proof.
Proof:
Statements
1. ∠MLN ≅ ∠PLO, ∠MNL ≅ ∠POL
2. MO ≅ NP
3. MO = NP
4. NO = NO
5. MO − NO = NP − NO
6. MO − NO = MN and NP − NO = OP
7. MN = OP
8. ∆MLN ≅ ∆PLO
9. ML ≅ PL
10. ∆MLP is isosceles.
Reasons
1. Given
2. Given
3. Definition of congruent line segments
4. Reflexive Property of Equality
5. Subtraction Property of Equality
6. Segment Addition Postulate
7. Substitution Property of Equality
8. [1]
9. [2]
10. Definition of isosceles triangle
a.
c.
b.
[1] ASA
[2] CPCTC
[1] CPCTC
[2] AAS
d.
8
[1] AAS
[2] CPCTC
[1] CPCTC
[2] ASA
Name: ________________________
ID: A
____ 18. In a swimming pool, two lanes are represented by lines l and m. If a string of flags strung across the lanes is
represented by transversal t, and x = 10, show that the lanes are parallel.
a.
b.
c.
d.
3x + 4 = 3(10) + 4 = 34°;
4x − 6 = 4(10) − 6 = 34°
The angles are same-side interior angles and they are supplementary, so the lanes are
parallel by the Converse of the Same-Side Interior Angles Theorem.
3x + 4 = 3(10) + 4 = 34°;
4x − 6 = 4(10) − 6 = 34°
The angles are alternate interior angles, and they are congruent, so the lanes are parallel
by the Converse of the Alternate Interior Angles Theorem.
3x + 4 = 3(10) + 4 = 34°;
4x − 6 = 4(10) − 6 = 34°
The angles are alternate interior angles and they are congruent, so the lanes are parallel
by the Alternate Interior Angles Theorem.
3x + 4 = 3(10) + 4 = 34°;
4x − 6 = 4(10) − 6 = 34°
The angles are corresponding angles and they are congruent, so the lanes are parallel by
the Converse of the Corresponding Angles Postulate.
9
Name: ________________________
ID: A
____ 19. Find CA.
a.
b.
c.
d.
CA = 10
Not enough information. An equiangular triangle is not necessarily equilateral.
CA = 12
CA = 14
10
Name: ________________________
ID: A
____ 20. Fill in the blanks to complete the two-column proof.
Given: ∠1 and ∠2 are supplementary. m∠1 = 135°
Prove: m∠2 = 45°
Proof:
Statements
1. ∠1 and ∠2 are supplementary.
2. [1]
3. m∠1 + m∠2 = 180°
4. 135° + m∠2 = 180°
5. m∠2 = 45°
a.
b.
c.
d.
Reasons
1. Given
2. Given
3. [2]
4. Substitution Property
5. [3]
[1] m∠1 = 135°
[2] Definition of supplementary angles
[3] Subtraction Property of Equality
[1] m∠1 = 135°
[2] Definition of supplementary angles
[3] Substitution Property
[1] m∠1 = 135°
[2] Definition of complementary angles
[3] Subtraction Property of Equality
[1] m∠2 = 135°
[2] Definition of supplementary angles
[3] Subtraction Property of Equality
11
Name: ________________________
ID: A
____ 21. Write a justification for each step, given that EG = FH .
EG = FH
EG = EF + FG
FH = FG + GH
EF + FG = FG + GH
EF = GH
a.
b.
c.
d.
Given information
[1]
Segment Addition Postulate
[2]
Subtraction Property of Equality
[1] Substitution Property of Equality
[2] Transitive Property of Equality
[1] Segment Addition Postulate
[2] Definition of congruent segments
[1] Segment Addition Postulate
[2] Substitution Property of Equality
[1] Angle Addition Postulate
[2] Subtraction Property of Equality
12
Name: ________________________
ID: A
____ 22. Given: RT ⊥SU , ∠SRT ≅ ∠URT , RS ≅ RU . T is the midpoint of SU .
Prove: ∆RTS ≅ ∆RTU
Complete the proof.
Proof:
Statements
1. RT ⊥SU
2. ∠RTS and ∠RTU are right angles.
3. ∠RTS ≅ ∠RTU
4. ∠SRT ≅ ∠URT
5. ∠S ≅ ∠U
6. RS ≅ RU
7. T is the midpoint of SU .
8. ST ≅ UT
9. RT ≅ RT
10. ∆RTS ≅ ∆RTU
a.
b.
c.
d.
Reasons
1. Given
2. [1]
3. Right Angle Congruence Theorem
4. Given
5. [2]
6. Given
7. Given
8. Definition of midpoint
9. [3]
10. Definition of congruent triangles
[1] Definition of perpendicular lines
[2] Third Angles Theorem
[3] Reflexive Property of Congruence
[1] Definition of perpendicular lines
[2] Third Angles Theorem
[3] Symmetric Property of Congruence
[1] Definition of right angles
[2] Third Angles Theorem
[3] Transitive Property of Congruence
[1] Definition of perpendicular lines
[2] Vertical Angles Theorem
[3] Symmetric Property of Congruence
13
Name: ________________________
ID: A
____ 23. Tell whether the transformation appears to be a reflection. Explain.
a.
b.
No; the image does not appear to be flipped.
Yes; the image appears to be flipped across a line.
____ 24. Find m∠RST .
a.
b.
m∠RST = 156°
m∠RST = 24°
c.
d.
m∠RST = 108°
m∠RST = 72°
____ 25. Find m∠1 in the diagram. (Hint: Draw a line parallel to the given parallel lines.)
a.
b.
m∠1 = 80°
m∠1 = 95°
c.
d.
14
m∠1 = 75°
m∠1 = 85°
Name: ________________________
ID: A
____ 26. A billiard ball bounces off the sides of a rectangular billiards table in such a way that ∠1 ≅ ∠3, ∠4 ≅ ∠6, and
∠3 and ∠4 are complementary. If m∠1 = 26.5°, find m∠3, m∠4, and m∠5.
a.
b.
c.
d.
m∠3 = 63.5°; m∠4 = 26.5°; m∠5 = 53°
m∠3 = 26.5°; m∠4 = 153.5°; m∠5 = 26.5°
m∠3 = 26.5°; m∠4 = 63.5°; m∠5 = 63.5°
m∠3 = 26.5°; m∠4 = 63.5°; m∠5 = 53°
15
Name: ________________________
ID: A
____ 27. Use the given two-column proof to write a flowchart proof.
Given: ∠1 ≅ ∠4
Prove: m∠2 = m∠3
Two-column proof:
Statements
1. ∠1 ≅ ∠4
2. ∠1 and ∠2 are supplementary. ∠3 and ∠4
are supplementary.
3. ∠2 ≅ ∠3
4. m∠2 = m∠3
Reasons
1. Given
2. Definition of linear pair
3. Congruent Supplements Theorem
4. Definition of congruent segments
Complete the proof.
Flowchart proof:
∠1 ≅ ∠4
Given
[1]
∠2 ≅ ∠3
Definition of linear pair
a.
b.
c.
d.
[2]
[1] ∠2 ≅ ∠3
[2] Definition of congruent segments
[1] ∠1 and ∠2 are supplements; ∠3 and ∠4 are supplementary
[2] Congruent Complements Theorem
[1] ∠1 and ∠2 are supplementary; ∠3 and ∠4 are supplementary
[2] Congruent Supplements Theorem
[1] Definition of congruent segments
[2] Congruent Supplements Theorem
16
m∠2 = m∠3
Definition of
congruent segments
Name: ________________________
ID: A
____ 28. Find m∠1 in the diagram. (Hint: Draw a line parallel to the given parallel lines.)
a.
b.
m∠1 = 120°
m∠1 = 130°
c.
d.
m∠1 = 125°
m∠1 = 135°
____ 29. The point G(4, 8) is rotated 90° about point M(−7, − 9) and then reflected across the line y = −6 . Find the
coordinates of the image G ′.
c. (−18, − 20)
a. (−24, − 14)
b. (12, − 14)
d. (−8, − 16)
17
Name: ________________________
ID: A
____ 30. Given: P is the midpoint of TQ and RS .
Prove: ∆TPR ≅ ∆QPS
Complete the proof.
Proof:
Statements
1. P is the midpoint of TQ and RS .
Reasons
1. Given
2. TP ≅ QP , RP ≅ SP
3. [2]
4. ∆TPR ≅ ∆QPS
2. [1]
a.
c.
b.
3. Vertical Angles Theorem
4. [3]
[1] Definition of midpoint
[2] RT ≅ SQ
[3] SSS
[1]. Definition of midpoint
[2] ∠TPR ≅ ∠QPS
[3] SAS
d.


→


→
____ 31. Draw and label a pair of opposite rays FG and FH .
a.
c.
b.
d.
18
[1] Definition of midpoint
[2] ∠TPR ≅ ∠QPS
[3] SSS
[1] Definition of midpoint
[2] ∠PRT ≅ ∠PSQ
[3] SAS
Name: ________________________
ID: A
____ 32. Using the information about John, Jason, and Julie, can you uniquely determine how they stand with respect
to each other? On what basis?
Statement 1: John and Jason are standing 12 feet apart.
Statement 2: The angle from Julie to John to Jason measures 31º.
Statement 3: The angle from John to Jason to Julie measures 49º.
a.
b.
c.
d.
Yes. They form a unique triangle by SAS.
No. There is no unique configuration.
Yes. They form a unique triangle by ASA.
Yes. They form a unique triangle by SSS.
____ 33. Find the measure of each exterior angle of a regular decagon.
a. 18°
c. 22.5°
b. 36°
d. 45°
____ 34. Tell whether the transformation appears to be a translation. Explain.
a.
b.
Yes; all of the points have moved the same distance in the same direction.
No; not all of the points have moved the same distance.
____ 35. Give an example of corresponding angles.
a.
b.
∠3 and ∠6
∠4 and ∠1
c.
d.
19
∠5 and ∠7
∠8 and ∠4
Name: ________________________
ID: A
____ 36. Tell whether the transformation appears to be a rotation. Explain.
a.
b.
Yes; the figure appears to be turned around a point.
No; the figure appears to be flipped.
20
Name: ________________________
ID: A
____ 37. Given: ∠CBF ≅ ∠CDG, AC bisects ∠BAD
Prove: AD ≅ AB
Complete the flowchart proof.
Proof:
∠CBF ≅ ∠CDG
∠ABC ≅ ∠ADC
Given
1.____________
AC bisects
∠BAD
Given.
2.____________
∆ACB ≅ ∆ACD
AD ≅ AB
Definition of
angle bisector.
4.___________
5._________
AC ≅ AC
3.____________
a.
b.
1. Congruent Supplements Theorem
2. ∠CAB ≅ ∠CAD
3. Reflexive Property of Congruence
4. AAS
5. CPCTC
1. Congruent Complements Theorem
2. ∠ACB ≅ ∠ACD
3. Transitive Property of Congruence
4. CPCTC
5. AAS
c.
d.
21
1. Congruent Complements Theorem
2. ∠ACB ≅ ∠ACD
3. Reflexive Property of Congruence
4. CPCTC
5. AAS
1. Congruent Supplements Theorem
2. ∠CAB ≅ ∠CAD
3. Transitive Property of Congruence
4. AAS
5. CPCTC
Name: ________________________
ID: A
____ 38. Use AAS to prove the triangles congruent.
←
→
←
→
Given: AB Ä GH , AC Ä FH , AC ≅ FH
Prove: ∆ABC ≅ ∆HGF
Complete the flowchart proof.
Proof:
AB Ä GH
∠B ≅ ∠G
Given
1.___________
←
→
←
→
AC Ä FH
Given
∠ACB ≅ ∠HFG
∆ABC ≅ ∆HGF
2.___________
AAS
AC ≅ FH
Given
a.
b.
c.
d.
1. Alternate Interior Angles Theorem
2. Alternate Exterior Angles Theorem
1. Alternate Interior Angles Theorem
2. Alternate Interior Angles Theorem
1. Alternate Exterior Angles Theorem
2. Alternate Interior Angles Theorem
1. Alternate Exterior Angles Theorem
2. Alternate Exterior Angles Theorem
22
Name: ________________________
ID: A
____ 39. Rotate ∆RSQ with vertices R(4, –1), S(5, 3), and Q(3, 1) by 90° about the origin.
a.
c.
b.
d.
____ 40. Find the measure of each interior angle of a regular 45-gon.
a. 188°
c. 172°
b. 176°
d. 164°
23
Name: ________________________
ID: A
____ 41. Write a two-column proof.
Given: t ⊥ l, ∠1 ≅ ∠2
Prove: m Ä l
Complete the proof.
Proof:
Statements
1. [1]
2. t ⊥ m
3. m Ä l
a.
b.
c.
d.
Reasons
1. Given
2. [2]
3. [3]
[1] t ⊥ l, ∠1 ≅ ∠2
[2] 2 intersecting lines form linear pair of ≅ ∠s → lines ⊥.
[3] Perpendicular Transversal Theorem
[1] t ⊥ l, ∠1 ≅ ∠2
[2] 2 intersecting lines form linear pair of ≅ ∠s → lines ⊥.
[3] 2 lines ⊥ to the same line → lines Ä.
[1] t ⊥ l, ∠1 ≅ ∠2
[2] 2 lines ⊥ to the same line → lines Ä.
[3] 2 intersecting lines form linear pair of ≅ ∠s → lines ⊥.
[1] t ⊥ l, ∠1 ≅ ∠2
[2] Perpendicular Transversal Theorem
[3] 2 lines ⊥ to the same line → lines Ä.
24
Name: ________________________
ID: A
____ 42. Use the given flowchart proof to write a two-column proof of the statement AF ≅ FD .
Flowchart proof:
AB = CD;
BF = FC
Given
AB + BF =
FC + CD
Addition
Property of
Equality
AB + BF = AF
FC + CD = FD
Segment
Addition
Postulate
AF = FD
AF ≅ FD
Substitution
Definition of
congruent segments
Complete the proof.
Two-column proof:
Statements
1. AB = CD; BF = FC
2. [1]
3. [2]
4. AF = FD
5. AF ≅ FD
a.
b.
c.
d.
[1]
[2]
[1]
[2]
[1]
[2]
[1]
[2]
Reasons
1. Given
2. Addition Property of Equality
3. Segment Addition Postulate
4. Substitution
5. Definition of congruent segments
AB + BF = AF ; FC + CD = FD
AF = FD
AB = CD; BF = FC
AB + BF = FC + CD
AF = FD
AB + BF = FC + CD
AB + BF = FC + CD
AB + BF = AF ;FC + CD = FD
25
Name: ________________________
ID: A
____ 43. Identify the transversal and classify the angle pair ∠11 and ∠7.
a.
b.
c.
d.
The transversal is line m. The angles are corresponding angles.
The transversal is line l. The angles are corresponding angles.
The transversal is line l. The angles are alternate interior angles.
The transversal is line n. The angles are alternate exterior angles.
____ 44. Show ∆ABD ≅ ∆CDB for a = 3.
Complete the proof.
AB = a + 7 = [1] = 10
CD = 4a − 2 = [2] = 12 − 2 = 10
AD = 6a − 2 = 6(3) − 2 = 18 − 2 = [3]
CB = [4]
AB ≅ CD. AD ≅ CB. BD ≅ BD by the Reflexive Property of Congruence. So ∆ABD ≅ ∆CDB by [5].
a.
[1] 3 + 7
[2] 4(3) − 2
[3] 16
[4] 16
[5] SSS
b.
[1] a + 7
[2] 4a − 2
[3] 16
[4] 16
[5] SAS
c.
26
[1] 3 + 7
[2] 4(3) − 2
[3] 26
[4] 26
[5] SSS
d.
[1] 3 + 7
[2] 4(3) − 2
[3] 16
[4] 16
[5] SAS
Name: ________________________
ID: A
____ 45. Tell whether ∠FAC and ∠3 are only adjacent, adjacent and form a linear pair, or not adjacent.
a.
b.
c.
only adjacent
not adjacent
adjacent and form a linear pair
____ 46. Find m∠E and m∠N , given m∠F = m∠P , m∠E = (x 2 )° , and m∠N = (4x 2 − 75)° .
a.
b.
m∠E = 65° , m∠N = 65°
m∠E = 25° , m∠N = 65°
c.
d.
m∠E = 65° , m∠N = 25°
m∠E = 25° , m∠N = 25°
____ 47. D is between C and E. CE = 6x, CD = 4x + 8, and DE = 27. Find CE.
a.
b.
CE = 17.5
CE = 105
c.
d.
27
CE = 57
CE = 78
Name: ________________________
ID: A
____ 48. Tell whether the transformation appears to be a translation. Explain.
a.
b.
Yes; all of the points have moved the same distance in the same direction.
No; not all of the points have moved the same distance.
____ 49. Find m∠ABC .
a.
b.
m∠ABC = 45º
m∠ABC = 50º
c.
d.
28
m∠ABC = 40º
m∠ABC = 35º
Name: ________________________
ID: A
____ 50. Use the given plan to write a two-column proof.
Given: m∠1 + m∠2 = 90°, m∠3 + m∠4 = 90°, m∠2 = m∠3
Prove: m∠1 = m∠4
Plan: Since both pairs of angle measures add to 90°, use substitution to show that the sums of both pairs are
equal. Since m∠2 = m∠3, use substitution again to show that sums of the other pairs are equal. Use the
Subtraction Property of Equality to conclude that m∠1 = m∠4.
Complete the proof.
Proof:
Statements
1. m∠1 + m∠2 = 90°
2. [1]
3. m∠1 + m∠2 = m∠3 + m∠4
4. m∠2 = m∠3
5. m∠1 + m∠2 = m∠2 + m∠4
6. m∠1 = m∠4
a.
b.
c.
d.
Reasons
1. Given
2. Given
3. Substitution Property
4. Given
5. [2]
6. [3]
[1] m∠3 + m∠4 = 90°
[2] Subtraction Property of Equality
[3] Substitution Property
[1] m∠3 + m∠4 = 90°
[2] Substitution Property
[3] Subtraction Property of Equality
[1] m∠5 + m∠6 = 90°
[2] Addition Property of Equality
[3] Substitution Property
[1] m∠5 + m∠6 = 90°
[2] Substitution Property
[3] Subtraction Property of Equality
29
ID: A
Geo Final Review 2014
Answer Section
MULTIPLE CHOICE
1. ANS: C
Let m∠A = x°. Then m∠B = (90 − x)°.
m∠A = 3m∠B + 2
x = 3(90 − x) + 2
x = 270 − 3x + 2
x = 272 − 3x
4x = 272
272
x= 4
x = 68
Substitute.
Distribute.
Combine like terms.
Add 3x to both sides.
Divide both sides by 4.
Simplify.
The measure of ∠A is 68°, so its complement is 22°.
Feedback
A
B
C
D
Check your equation. The original angle is 2 degrees more than 3 times its complement.
Simplify the terms when solving.
Correct!
This is the original angle. Find the measure of the complement.
PTS:
OBJ:
NAT:
KEY:
1
DIF: Average
REF: Page 29
1-4.3 Using Complements and Supplements to Solve Problems
12.3.3.g
STA: 6MG2.2
TOP: 1-4 Pairs of Angles
complementary angles | supplementary angles
1
ID: A
2. ANS: C
Step 1 Solve for x.
m∠ABD = m∠DBC
Definition of angle bisector.
(7x − 1)° = (4x + 8)°
Substitute 7x − 1 for ∠ABD and 4x + 8 for ∠DBC .
7x = 4x + 9
3x = 9
x=3
Add 1 to both sides.
Subtract 4x from both sides.
Divide both sides by 3.
Step 2 Find m∠ABD.
m∠ABD = 7x − 1 = 7(3) − 1 = 20°
Feedback
A
B
C
D
Check your simplification technique.
Substitute this value of x into the expression for the angle.
Correct!
This answer is the entire angle. Divide by two.
PTS: 1
DIF: Average
REF: Page 23
OBJ: 1-3.4 Finding the Measure of an Angle
NAT: 12.2.1.f
STA: GE1.0
TOP: 1-3 Measuring and Constructing Angles
KEY: angle bisectors | angle measures
3. ANS: C
m∠1 = 3(20) + 30 = 90° ;
Substitute 20 for x.
m∠2 = 5(20) − 10 = 90°
Substitution Property of Equality
m∠1 = m∠2 = 90°
lÄm
Converse of the Alternate Interior Angles Theorem
Feedback
A
B
C
D
Angles 1 and 2 are alternate interior angles and are congruent.
Angles 1 and 2 are alternate interior angles and are congruent.
Correct!
Angles 1 and 2 are alternate interior angles and are congruent.
PTS: 1
DIF: Average
REF: Page 164
OBJ: 3-3.2 Determining Whether Lines are Parallel
STA: GE7.0
TOP: 3-3 Proving Lines Parallel
2
NAT: 12.3.3.g
ID: A
4. ANS: C
The complete translation is (x, y) → (x, y) + 6 ⋅ (0, 4) = (x, y) + (0, 24) = (x + 0, y + 24) = (x, y + 24).
A(2, 0) → A(2, 0 + 24) = A′(2, 24)
B(0, 2) → B(0, 2 + 24) = B ′(0, 26)
C(−2, 0) → C(−2, 0 + 24) = C ′(−2, 24)
D(0, − 2) → D(0, − 2 + 24) = D ′(0, 22)
Feedback
A
B
C
D
Add 24 to each y-coordinate.
Add 24 to each y-coordinate.
Correct!
Add 24 to each y-coordinate.
PTS: 1
DIF: Average
REF: Page 833
OBJ: 12-2.4 Application
NAT: 12.3.2.c
STA: GE22.0
TOP: 12-2 Translations
5. ANS: A
The perimeter is 33 units and it is an equilateral triangle, so each side has length 11 units.
Use this to solve for either side.
11 = 2y + 3
11 = y 2 − 5
8 = 2y
16 = y 2
4=y
4=y
An answer of −4 does not apply here.
Feedback
A
B
C
D
Correct!
This is the length of each side. Now find the value of y.
When solving 2y + 3 = 11, subtract 3 from both sides of the equation.
The perimeter is 33 so the length of each side is 11. Set one of the sides equal to 11 and
solve for y.
PTS: 1
DIF: Advanced
TOP: 4-1 Classifying Triangles
NAT: 12.2.1.h
3
STA: GE12.0
ID: A
6. ANS: A
Two-column proof:
Statements
1. ∠BAC is a right angle. ∠1 ≅ ∠3
2. m∠BAC = 90°
3. m∠BAC = m∠1 + m∠2
4. m∠1 + m∠2 = 90°
5. m∠1 = m∠3
6. m∠3 + m∠2 = 90°
7. ∠2 and ∠3 are complementary.
Reasons
1. Given
2. Definition of a right angle
3. Angle Addition Postulate
4. Substitution
5. Definition of congruent angles
6. Substitution
7. Definition of complementary angles
Feedback
A
B
C
D
Correct!
In a paragraph proof, statements and reasons appear together.
In a paragraph proof, statements and reasons appear together.
In a paragraph proof, statements and reasons appear together.
PTS: 1
DIF: Average
NAT: 12.3.5.a
STA: GE2.0
7. ANS: A
Subtract from 90º and simplify.
90° − 31.1°= 58.9°
REF: Page 120
OBJ: 2-7.3 Reading a Paragraph Proof
TOP: 2-7 Flowchart and Paragraph Proofs
Feedback
A
B
C
D
Correct!
Find the measure of a complementary angle, not a supplementary angle.
The measures of complementary angles add to 90 degrees.
Complementary angles are angles whose measures have a sum of 90 degrees.
PTS:
OBJ:
NAT:
KEY:
1
DIF: Basic
REF: Page 29
1-4.2 Finding the Measures of Complements and Supplements
12.3.3.g
STA: 6MG2.2
TOP: 1-4 Pairs of Angles
complementary angles | supplementary angles
4
ID: A
8. ANS: D
Let the acute angles be ∠M and ∠N , with m∠M = 34.6°.
m∠M + m∠N = 90°
The acute angles of a right triangle are complementary.
Substitute 34.6° for m∠M .
34.6° + m∠N = 90°
m∠N = 55.4°
Subtract 34.6° from both sides.
Feedback
A
B
C
D
The measure of the other acute angle is less than 90 degrees.
The two acute angles in a right triangle are complementary.
This is the measure of the given angle. Find the measure of the other acute angle.
Correct!
PTS: 1
DIF: Basic
REF: Page 225
OBJ: 4-2.2 Finding Angle Measures in Right Triangles
NAT: 12.3.3.f
STA: GE12.0
TOP: 4-2 Angle Relationships in Triangles
5
ID: A
9. ANS: A
Step 1 Draw two lines m, n, and a transversal
p such that ∠1 and ∠2 are alternate interior
angles. They should lie on opposite sides of
the transversal p between lines m and n.
Step 3 ∠3 and ∠4 are alternate exterior
angles. They should lie on opposite sides of
the transversal p and outside lines m and n.
Add ∠4 to the drawing.
Step 2 ∠2 and ∠3 are corresponding angles.
Corresponding angles lie on the same side of
the transversal p and on the same sides of
lines m and n. Add ∠3 to the drawing.
∠1 and ∠4 are corresponding angles. They
lie on the same side of the transversal p and
on the same sides of lines m and n.
Feedback
A
B
C
D
Correct!
Angles 2 and 3 are corresponding angles and should lie on the same side of transversal
p, on the same sides of lines m and n.
Angles 2 and 3 are corresponding angles and should lie on the same side of transversal
p, on the same sides of lines m and n.
Angles 1 and 2 are alternate interior angles and should lie on opposite sides of
transversal p, between lines m and n.
PTS: 1
DIF: Advanced
TOP: 3-1 Lines and Angles
NAT: 12.2.1.f
KEY: multi-step
6
STA: GE7.0
ID: A
10. ANS: B
m∠JKL = m∠JKM + m∠MKL
100° = (6x + 8)° + (2x − 4)°
100 = 8x + 4
96 = 8x
12 = x
x = 12
[1] Angle Addition Postulate
Substitution Property of Equality
Simplify.
Subtraction Property of Equality
[2] Division Property of Equality
Symmetric Property of Equality
Feedback
A
B
C
D
Check the justifications.
Correct!
The Segment Addition Postulate refers to segments, not angles.
Check the properties.
PTS: 1
DIF: Average
REF: Page 106
OBJ: 2-5.3 Solving an Equation in Geometry
NAT: 12.5.2.e
STA: GE1.0
TOP: 2-5 Algebraic Proof
11. ANS: A
It is given that ST ≅ UT . Since ∠RTS and ∠RTU are right angles, ∠RTS ≅ ∠RTU by the Right Angle
Congruence Theorem. By the Reflexive Property of Congruence, RT ≅ RT . Therefore, ∆RTS ≅ ∆RTU by
SAS.
Feedback
A
B
C
D
Correct!
Segment SU being congruent to itself does not help in proving the triangles congruent.
Angle SRT and angle URT are not right angles.
Check the figure to see what is given.
PTS: 1
DIF: Average
REF: Page 243
OBJ: 4-4.2 Application
NAT: 12.3.5.a
STA: GE5.0
TOP: 4-4 Triangle Congruence: SSS and SAS
12. ANS: A
A polygon is a closed plane figure formed by three or more segments that intersect only at their endpoints. A
polygon with 10 sides is called a decagon.
Feedback
A
B
C
D
Correct!
A hexagon has 6 sides.
A dodecagon has 12 sides.
A polygon is a closed plane figure formed by three or more segments that intersect only
at their endpoints.
PTS: 1
NAT: 12.3.3.f
DIF: Basic
STA: GE12.0
REF: Page 382
OBJ: 6-1.1 Identifying Polygons
TOP: 6-1 Properties and Attributes of Polygons
7
ID: A
13. ANS: D
∠1 ≅ ∠2 is given. From the diagram, ∠1 and ∠2 are corresponding angles. So by the Converse of the
Corresponding Angles Postulate, l Ä m.
Feedback
A
B
C
D
Use the given information.
Use the Converse of the Corresponding Angles Postulate.
Use the Converse of the Corresponding Angles Postulate.
Correct!
PTS: 1
DIF: Basic
REF: Page 162
OBJ: 3-3.1 Using the converse of the Corresponding Angles Postulate
NAT: 12.3.3.g
STA: GE7.0
TOP: 3-3 Proving Lines Parallel
14. ANS: D
The Third Angles Theorem states that if two angles of one triangle are congruent to two angles of another
triangle, then the third pair of angles are congruent.
It is given that ∠A ≅ ∠F and ∠B ≅ ∠E . Therefore, ∠CDE ≅ ∠DCB. So, m∠DCB = 46°.
Feedback
A
B
C
D
This is the complement. Use the Third Angles Theorem.
This is the supplement. Use the Third Angles Theorem.
The Third Angles Theorem states that if two angles of one triangle are congruent to two
angles of another triangle, then the third pair of angles are congruent.
Correct!
PTS: 1
DIF: Advanced
NAT: 12.3.3.f
STA: GE12.0
TOP: 4-2 Angle Relationships in Triangles
15. ANS: A
AC ≅ DC is given. ∠CAB ≅ ∠CDE because both are right angles. By the Vertical Angles Theorem,
∠ACB ≅ ∠DCE . Therefore, ∆CBA ≅ ∆CED by ASA.
Feedback
A
B
C
D
Correct!
Adjacent angles are angles in a plane that have their vertex and one side in common but
have no interior points in common. Angle ACB and angle DCE are not adjacent angles.
Use ASA, not SAS, to prove the triangles congruent.
Look for vertical angles.
PTS: 1
NAT: 12.3.2.e
DIF: Basic
STA: GE5.0
REF: Page 253
OBJ: 4-5.2 Applying ASA Congruence
TOP: 4-5 Triangle Congruence: ASA AAS and HL
8
ID: A
16. ANS: A
The image of (x,y) is (x − 1, y + 3).
A(3, 4) → A(3 − 1, 4 + 3) = A′(1, 7)
B(2, − 1) → B(2 − 1, − 1 + 3) = B ′(1, 2)
C(4, 12) → C(4 − 1, 12 + 3) = C ′(3, 15)
Feedback
A
B
C
D
Correct!
Subtract 1 from each x-coordinate. Add 3 to each y-coordinate.
Subtract 1 from each x-coordinate. Add 3 to each y-coordinate.
Subtract 1 from each x-coordinate. Add 3 to each y-coordinate.
PTS: 1
DIF: Average
REF: Page 833
OBJ: 12-2.3 Drawing Translations in the Coordinate Plane
NAT: 12.3.2.c
STA: GE22.0
TOP: 12-2 Translations
17. ANS: C
[1] Steps 1 and 7 state that two angles and a nonincluded side of ∆MLN and ∆PLO are congruent. By AAS,
∆MLN ≅ ∆PLO.
[2] Since ∆MLN ≅ ∆PLO, by CPCTC, ML ≅ PL .
Feedback
A
B
C
D
Steps 1 and 7 state that two angles and a nonincluded side of triangle MLN and triangle
PLO are congruent. Which triangle congruence theorem states that the triangles are
congruent?
Before using CPCTC, you must prove that triangle MLN and triangle PLO are
congruent.
Correct!
Before using CPCTC, you must prove that triangle MLN and triangle PLO are
congruent. Since steps 1 and 7 state that two angles and a nonincluded side are
congruent, which triangle congruence theorem states that the triangles are congruent?
PTS: 1
NAT: 12.3.5.a
DIF: Average
STA: GE5.0
REF: Page 261
OBJ: 4-6.3 Using CPCTC in a Proof
TOP: 4-6 Triangle Congruence: CPCTC
9
ID: A
18. ANS: B
Substitute 10 for x in each expression:
3x + 4 = 3(10) + 4 = 34°
4x − 6 = 4(10) − 6 = 34°
The angles are alternate interior angles, and they are congruent, so the lanes are parallel by the Converse of
the Alternate Interior Angles Theorem.
Feedback
A
B
C
D
The angles are alternate interior angles.
Correct!
The lanes are parallel by the Converse of the Alternate Interior Angles Theorem.
The angles are alternate interior angles.
PTS: 1
DIF: Average
REF: Page 165
OBJ: 3-3.4 Application
NAT: 12.3.5.a
STA: GE7.0
TOP: 3-3 Proving Lines Parallel
19. ANS: D
Equiangular triangles are equilateral.
∆ABC is equilateral.
2s − 10 = s + 2
Definition of equilateral triangle.
s = 12
Subtract s and add 10 to both sides of the equation.
AB = 2s − 10
AB = 2(12) − 10
AB = 14
Substitute 12 for s in the equation for AB.
Simplify.
CA = AB
CA = 14
Definition of equilateral triangle.
Substitute 14 for AB.
Feedback
A
B
C
D
Equiangular triangles are equilateral. Use AB = BC to solve for s, and then use AC = AB
or AC = BC to find AC.
By a corollary to the Isosceles Triangle Theorem, equiangular triangles are equilateral.
Use AB = BC to solve for s, and then use AC = AB or AC = BC to find AC.
This is s. Substitute s in the original equation to find AC.
Correct!
PTS: 1
DIF: Basic
REF: Page 275
OBJ: 4-8.3 Using Properties of Equilateral Triangles
NAT: 12.3.3.f
STA: GE12.0
TOP: 4-8 Isosceles and Equilateral Triangles
10
ID: A
20. ANS: A
Proof:
Statements
1. ∠1 and ∠2 are supplementary.
2. m∠1 = 135°
3. m∠1 + m∠2 = 180°
4. 135° + m∠2 = 180°
5. m∠2 = 45°
Reasons
1. Given
2. Given
3. Definition of supplementary angles
4. Substitution Property
5. Subtraction Property of Equality
Feedback
A
B
C
D
Correct!
To get from Step 4 to Step 5, use subtraction, not substitution.
The angles are supplementary, not complementary.
Check to the given information.
PTS: 1
DIF: Average
REF: Page 111
OBJ: 2-6.2 Completing a Two-Column Proof
NAT: 12.3.5.a
STA: GE2.0
TOP: 2-6 Geometric Proof
21. ANS: C
EG = FH
Given information
EG = EF + FG
Segment Addition Postulate
FH = FG + GH
Segment Addition Postulate
EF + FG = FG + GH
Substitution Property of Equality
EF = GH
Subtraction Property of Equality
Feedback
A
B
C
D
Check the properties.
Check the steps.
Correct!
The Angle Addition Postulate refers to angles, not segments.
PTS: 1
NAT: 12.3.5.a
DIF: Average
STA: GE2.0
REF: Page 110
OBJ: 2-6.1 Writing Justifications
TOP: 2-6 Geometric Proof
11
ID: A
22. ANS: A
Proof:
Statements
1. RT ⊥SU
2. ∠RTS and ∠RTU are right angles.
3. ∠RTS ≅ ∠RTU
4. ∠SRT ≅ ∠URT
5. ∠S ≅ ∠U
6. RS ≅ RU
7. T is the midpoint of SU .
8. ST ≅ UT
9. RT ≅ RT
10. ∆RTS ≅ ∆RTU
Reasons
1. Given
2. Definition of perpendicular lines
3. Right Angle Congruence Theorem
4. Given
5. Third Angles Theorem
6. Given
7. Given
8. Definition of midpoint
9. Reflexive Property of Congruence
10. Definition of congruent triangles
Feedback
A
B
C
D
Correct!
Use the correct property to show that the part is congruent to itself.
Use the definition of perpendicular lines to show that the lines intersect to form right
angles.
Angle S and angle U are not vertical angles. Use a different justification for Reason 5.
PTS: 1
DIF: Average
REF: Page 232
OBJ: 4-3.3 Proving Triangles Congruent
NAT: 12.3.5.a
STA: GE5.0
TOP: 4-3 Congruent Triangles
23. ANS: B
A reflection is a transformation that moves a figure (the preimage) by flipping it across a line.
Feedback
A
B
See if you can flip the image across the line to get a congruent image.
Correct!
PTS: 1
NAT: 12.3.2.c
DIF: Basic
STA: GE22.0
REF: Page 824
OBJ: 12-1.1 Identifying Reflections
TOP: 12-1 Reflections
12
ID: A
24. ANS: D
(3x)° = (4x − 24)°
−x = −24
x = 24
m∠RST = 3x = 3(24) = 72°
Alternate Exterior Angles Theorem
Subtract 4x from both sides.
Divide both sides by −1.
Substitute 24 for x .
Feedback
A
B
C
D
After finding x, substitute to find the angle measure.
Find the measure of angle RST, not the value of x.
Find the measure of angle RST, not the supplement.
Correct!
PTS: 1
DIF: Average
REF: Page 156
OBJ: 3-2.2 Finding Angle Measures
NAT: 12.3.3.g
STA: GE7.0
TOP: 3-2 Angles Formed by Parallel Lines and Transversals
25. ANS: D
Step 1 Draw line l parallel to lines m and n.
Step 2 Find m∠x.
m∠1 = m∠x + m∠y
Use the Corresponding Angles Postulate with
lines m and l. m∠x = 35°.
Step 3 Find m∠y.
Use the Same-Side Interior Angles Theorem
with lines l and n. m∠y = 180 − 130 = 50°.
Step 4 Find m∠1.
m∠1 = m∠x + m∠y = 35 + 50 = 85°
Feedback
A
B
C
D
Use the Corresponding Angles Postulate and a theorem related to parallel lines and
angle pairs.
Use the Corresponding Angles Postulate and a theorem related to parallel lines and
angle pairs.
Use the Corresponding Angles Postulate and a theorem related to parallel lines and
angle pairs.
Correct!
PTS: 1
DIF: Advanced
NAT: 12.2.1.f
TOP: 3-2 Angles Formed by Parallel Lines and Transversals
13
STA: GE7.0
KEY: multi-step
ID: A
26. ANS: D
Since ∠1 ≅ ∠3, m∠1 ≅ m∠3 .
Thus m∠3 = 26.5°.
Since ∠3 and ∠4 are complementary,
m∠4 = 90° − 26.5° = 63.5°.
Since ∠4 ≅ ∠6, m∠4 ≅ m∠6 .
Thus m∠6 = 63.5°.
By the Angle Addition Postulate,
180° = m∠4 + m∠5 + m∠6
= 63.5° + m∠5 + 63.5°
Thus, m∠5 = 53°.
Feedback
A
B
C
D
Angle 1 and angle 3 are congruent. Congruent angles have the same measure.
Angle 3 and angle 4 are complementary, not supplementary.
The measure of angle 5 is 180 degrees minus the sum of the measure of angle 4 and the
measure of angle 6.
Correct!
PTS: 1
DIF: Average
REF: Page 30
OBJ: 1-4.4 Problem-Solving Application
NAT: 12.3.3.g
STA: 6MG2.2
TOP: 1-4 Pairs of Angles
KEY: application | complementary angles | supplementary angles
27. ANS: C
In a flowchart, reasons follow statements. Using the two-column proof, the statement that leads to Reason 2
is ∠1 and ∠2 are supplementary; ∠3 and ∠4 are supplementary. The reason that follows Statement 3 is
Congruent Supplements Theorem.
Feedback
A
B
C
D
In a flowchart, reasons follow statements.
Angles 1 and 2 are supplements, not complements.
Correct!
In a flowchart, reasons follow statements.
PTS: 1
NAT: 12.3.5.a
DIF: Average
STA: GE2.0
REF: Page 119
OBJ: 2-7.2 Writing a Flowchart Proof
TOP: 2-7 Flowchart and Paragraph Proofs
14
ID: A
28. ANS: D
Step 1 Draw line l parallel to lines m and n. Step 2 Use the Alternate Interior Angles
Given: m∠y + m∠z = 90° , ∠x ≅ ∠w,
Theorem to find pairs of congruent angles.
∠y ≅ ∠x, ∠z ≅ ∠w
mÄ n Ä l
m∠y = m∠x , m∠z = m∠w
Step 3 Substitute x for y and w for z in the
given m∠y + m∠z = 90° .
m∠x + m∠w = 90°
Step 4 Use the definition of congruent angles
and the given ∠x ≅ ∠w.
m∠x = m∠w
Step 5 To find m∠w , substitute w for x.
m∠x + m∠w = 90°
m∠w + m∠w = 90°
2 ⋅ m∠w = 90°
m∠w = 45°
Step 6 Find m∠1.
∠1 and ∠w are supplementary.
m∠1 + m∠w = 180°
m∠1 + 45° = 180°
m∠1 = 135°
Feedback
A
B
C
D
Draw a line parallel to the given parallel lines and use the Alternate Interior Angles
Theorem.
Draw a line parallel to the given parallel lines and use the Alternate Interior Angles
Theorem.
Draw a line parallel to the given parallel lines and use the Alternate Interior Angles
Theorem.
Correct!
PTS: 1
DIF: Advanced
TOP: 3-4 Perpendicular Lines
NAT: 12.2.1.f
KEY: multi-step
15
STA: GE7.0
ID: A
29. ANS: A
Rotation 90° about the origin of any
point A(x, y) results in the image
A′(−y, x). To rotate the point about
h , the
M(−7, − 9), determine ä
horizontal vector, and ä
v , the vertical
vector, from G to M. Then move
ä
h vertically from M, and move the
opposite of ä
v horizontally from M.
The result of the rotation is labeled G ′
in the graph.
The line y = −6 is a horizontal line
passing through (0, − 6). Reflection
across a horizontal line involves
movement of the point to the other side
of the line, such that the image is the
same distance from the line that the
original point was. The x-coordinate
does not change. The result of the
reflection is labeled G ″ in the graph.
Feedback
A
B
C
D
Correct!
The point is reflected across a horizontal line, not a vertical line.
Rotation occurs before reflection.
Rotate around the given point, not around the origin.
PTS: 1
DIF: Advanced
NAT: 12.3.2.c
TOP: 12-4 Compositions of Transformations
16
STA: GE22.0
ID: A
30. ANS: B
Proof:
Statements
1. P is the midpoint of TQ and RS .
2. TP ≅ QP , RP ≅ SP
3. ∠TPR ≅ ∠QPS
4. ∆TPR ≅ ∆QPS
Reasons
1. Given
2. Definition of midpoint
3. Vertical Angles Theorem
4. SAS
Feedback
A
B
C
D
There is not enough information to show that segment RT is congruent to segment SQ.
Correct!
Use the correct postulate to prove the triangles congruent.
Angle PRT and angle PSQ are not vertical angles.
PTS: 1
NAT: 12.3.5.a
31. ANS: D
DIF: Average
STA: GE5.0


→
REF: Page 244
OBJ: 4-4.4 Proving Triangles Congruent
TOP: 4-4 Triangle Congruence: SSS and SAS


→
←

→
In the diagram, rays FG and FH share a common endpoint F and form the line GH .
Feedback
A
B
C
D
Opposite rays form a line.
Opposite rays are two rays that have a common endpoint and form a line.
Opposite rays form a line.
Correct!
PTS: 1
DIF: Basic
NAT: 12.3.1.d
STA: GE1.0
KEY: opposite rays
REF: Page 7
OBJ: 1-1.2 Drawing Segments and Rays
TOP: 1-1 Understanding Points Lines and Planes
17
ID: A
32. ANS: C
Statements 2 and 3 determine the measures of two angles of the triangle.
Statement 1 determines the length of the included side.
By ASA, the triangle must be unique.
Feedback
A
B
C
D
There is not enough information for SAS. Draw a diagram to help you.
Draw a diagram. There is enough information to determine a unique triangle.
Correct!
There is not enough information for SSS. Draw a diagram to help you.
PTS: 1
DIF: Average
REF: Page 252
OBJ: 4-5.1 Problem-Solving Application
NAT: 12.3.3.f
STA: 7MR3.1
TOP: 4-5 Triangle Congruence: ASA AAS and HL
33. ANS: B
A decagon has 10 sides and 10 vertices.
sum of exterior angle measures = 360°
Polygon Exterior Angle Sum Theorem
A regular decagon has 10 congruent exterior
360
measure of one exterior angle =
= 36°
angles, so divide the sum by 10.
10
The measure of each exterior angle of a regular decagon is 36°.
Feedback
A
B
C
D
Divide 360 by the number of sides.
Correct!
Divide 360 by the number of sides the polygon has.
Divide by the number of sides the polygon has.
PTS: 1
DIF: Average
REF: Page 384
OBJ: 6-1.4 Finding Exterior Angle Measures in Polygons
NAT: 12.3.3.f
STA: GE12.0
TOP: 6-1 Properties and Attributes of Polygons
18
ID: A
34. ANS: A
A translation is a transformation where all the points of a figure are moved the same distance in the same
direction.
This transformation is a translation because all of the points have moved the same distance in the same
direction.
Feedback
A
B
Correct!
Check where all the points have moved.
PTS: 1
DIF: Basic
REF: Page 831
OBJ: 12-2.1 Identifying Translations
NAT: 12.3.2.c
STA: GE22.0
TOP: 12-2 Translations
35. ANS: D
Corresponding angles lie on the same side of a transversal, on the same sides of the two lines the transversal
crosses. So, ∠8 and ∠4 are corresponding angles.
Feedback
A
B
C
D
Corresponding angles lie on the same side of a transversal, on the same sides of two
lines.
Angle 4 and angle 1 are supplementary angles, not corresponding angles.
Angle 5 and angle 7 are vertical angles, not corresponding angles.
Correct!
PTS: 1
NAT: 12.3.3.g
DIF: Basic
STA: GE7.0
REF: Page 147
OBJ: 3-1.2 Classifying Pairs of Angles
TOP: 3-1 Lines and Angles
19
ID: A
36. ANS: B
: This appears to be a reflection.
A rotation is a transformation that turns a figure around a fixed point, called the center of rotation. If the
transformation is a rotation, then the figure on the left rotates clockwise 90° about a fixed point to look like
this:
Feedback
A
B
See if you can rotate the image around a fixed point and get a congruent image.
Correct!
PTS: 1
DIF: Basic
REF: Page 839
OBJ: 12-3.1 Identifying Rotations
NAT: 12.3.2.c
STA: GE22.0
TOP: 12-3 Rotations
37. ANS: A
1a. By the Linear Pair Theorem, ∠CBF and ∠ABC are supplementary and ∠CDG and ∠ADC are
supplementary.
1b. Given ∠CBF ≅ ∠CDG, by the Congruent Supplements Theorem, ∠ABC ≅ ∠ADC .
2. ∠CAB ≅ ∠CAD by the definition of an angle bisector.
3. AC ≅ AC by the Reflexive Property of Congruence
4. Two angles and a nonincluded side of ∆ACB and ∆ACD are congruent. By AAS,
∆ACB ≅ ∆ACD.
5. Since ∆ACB ≅ ∆ACD, AD ≅ AB by CPCTC.
Feedback
A
B
C
D
Correct!
For reason 1, check whether the linear pairs are complementary or supplementary.
For statement 2, use the fact that line segment AC bisects angle A, not angle C.
Find the correct property that states that a line segment is congruent to itself.
PTS: 1
DIF: Average
REF: Page 260
OBJ: 4-6.2 Proving Corresponding Parts Congruent
STA: GE5.0
TOP: 4-6 Triangle Congruence: CPCTC
20
NAT: 12.3.5.a
ID: A
38. ANS: A
1. ∠B and ∠G are alternate interior angles and AB Ä GH . Thus by the Alternate Interior Angles Theorem,
∠B ≅ ∠G.
←
→
←
→
2. ∠ACB and ∠HFG are alternate exterior angles and AC Ä FH . Thus by the Alternate Exterior Angles
Theorem, ∠ACB ≅ ∠HFG.
Feedback
A
B
C
D
Correct!
If line AC is parallel to line FG, are angle ACB and angle HFG alternate interior angles
or alternate exterior angles?
You switched the definitions of alternate interior and alternate exterior angles.
If line segment AB is parallel to line segment GH, are angle B and angle G alternate
exterior angles or alternate interior angles?
PTS: 1
DIF: Average
REF: Page 254
OBJ: 4-5.3 Using AAS to Prove Triangles Congruent
NAT: 12.3.5.a
STA: GE5.0
TOP: 4-5 Triangle Congruence: ASA AAS and HL
39. ANS: C
The image of (x, y) is (–y, x).
R(4, –1) → R ′(1, 4)
S(5, 3) → S ′(–3, 5)
Q(3, 1) → Q ′(–1, 3)
Graph the preimage and the image.
Feedback
A
B
C
D
This is a rotation by 180° about the origin.
This is a reflection across the y-axis.
Correct!
The rotation is 90° counterclockwise, not clockwise.
PTS: 1
DIF: Average
REF: Page 841
OBJ: 12-3.3 Drawing Rotations in the Coordinate Plane
STA: GE22.0
TOP: 12-3 Rotations
21
NAT: 12.3.2.c
ID: A
40. ANS: C
Step 1 Find the sum of the interior angle measures.
(n – 2)180°
Polygon Angle Sum Theorem
= (45 – 2)180°
A 45-gon has 45 sides, so substitute 45 for n.
Simplify.
= 7740
Step 2 Find the measure of one interior angle.
7740
= 172
The interior angles are ≅ , so divide by 45.
45
Feedback
A
B
C
D
Subtract, not add, 2 from the number of sides.
Subtract 2, not 1, from the number of sides.
Correct!
According to the Polygon Angle Sum Theorem, the sum of the interior angle measures
is the product of 180 and the number of sides minus 2.
PTS: 1
DIF: Average
REF: Page 384
OBJ: 6-1.3 Finding Interior Angle Measures and Sums in Polygons
NAT: 12.3.3.f
STA: GE12.0
TOP: 6-1 Properties and Attributes of Polygons
41. ANS: B
Proof:
Statements
Reasons
1. Given
1. t ⊥ l, ∠1 ≅ ∠2
2. t ⊥ m
2. If 2 intersecting lines form linear pair of ≅
∠s → lines ⊥.
3. m Ä l
3. If 2 lines ⊥ to the same line → lines Ä.
Feedback
A
B
C
D
Reason 2 is if 2 intersecting lines form a linear pair of congruent angles, then the lines
are perpendicular. Reason 3 is if 2 lines are perpendicular to the same line, then the
lines are parallel.
Correct!
Switch Reason 2 and Reason 3.
Reason 2 is if 2 intersecting lines form a linear pair of congruent angles, then the lines
are perpendicular. Reason 3 is if 2 lines are perpendicular to the same line, then the
lines are parallel.
PTS: 1
NAT: 12.3.5.a
DIF: Basic
STA: GE2.0
REF: Page 173
OBJ: 3-4.2 Proving Properties of Lines
TOP: 3-4 Perpendicular Lines
22
ID: A
42. ANS: D
In a flowchart, reasons flow from the statement above. The statement above Reason 2 is AB + BF = FC + CD.
The statement above Reason 3 is AB + BF = AF ; FC + CD = FD.
Feedback
A
B
C
D
Reasons flow from the statement above.
Reasons flow from the statement above.
Reasons flow from the statement above.
Correct!
PTS: 1
DIF: Average
REF: Page 118
OBJ: 2-7.1 Reading a Flowchart Proof
NAT: 12.3.5.a
STA: GE2.0
TOP: 2-7 Flowchart and Paragraph Proofs
43. ANS: B
To determine which line is the transversal for a given angle pair, locate the line that connects the vertices.
Corresponding angles lie on the same side of the transversal l, on the same sides of lines n and m.
Feedback
A
B
C
D
To find which line is the transversal for a given angle pair, locate the line that connects
the vertices.
Correct!
Alternate interior angles lie on opposite sides of the transversal, between two lines.
To find which line is the transversal for a given angle pair, locate the line that connects
the vertices.
PTS: 1
DIF: Average
REF: Page 147
OBJ: 3-1.3 Identifying Angle Pairs and Transversals
STA: GE7.0
TOP: 3-1 Lines and Angles
44. ANS: A
AB = a + 7 = 3 + 7 = 10
CD = 4a − 2 = 4(3) − 2 = 12 − 2 = 10
AD = 6a − 2 = 6(3) − 2 = 18 − 2 = 16
CB = 16
NAT: 12.3.3.g
AB ≅ CD. AD ≅ CB. BD ≅ BD by the Reflexive Property of Congruence. So ∆ABD ≅ ∆CDB by SSS.
Feedback
A
B
C
D
Correct!
Substitute 3 for a.
Check the measures of segment AD and segment CB.
Use the correct postulate.
PTS: 1
DIF: Average
REF: Page 244
OBJ: 4-4.3 Verifying Triangle Congruence
NAT: 12.3.5.a
STA: GE2.0
TOP: 4-4 Triangle Congruence: SSS and SAS
23
ID: A
45. ANS: C


→


→
∠FAC and ∠3 are adjacent angles. Their noncommon sides, AF and AG , are opposite rays, so ∠FAC and
∠3 also form a linear pair.
Feedback
A
B
C
Adjacent angles form a linear pair if and only if their noncommon sides are opposite
rays.
Two angles are adjacent if they have a common vertex and a common side, but no
common interior points.
Correct!
PTS: 1
DIF: Average
REF: Page 28
OBJ: 1-4.1 Identifying Angle Pairs
NAT: 12.3.3.g
STA: 6MG2.1
TOP: 1-4 Pairs of Angles
KEY: angle pairs | linear pair | adjacent
46. ANS: D
∠E ≅ ∠N
Third Angles Theorem
m∠E = m∠N
Definition of congruent angles
2
2
(x )° = (4x − 75)°
Substitute x 2 for m∠E and 4x 2 − 75 for m∠N .
−3x 2 = −75
x 2 = 25
Subtract 4x 2 from both sides.
Divide both sides by –3.
So m∠E = 25° .
Since m∠E = m∠N , m∠N = 25° .
Feedback
A
B
C
D
These are the measures of angles F and P, not angles E and N.
Use the Third Angles Theorem.
The Third Angles Theorem states that if two angles of one triangle are congruent to two
angles of another triangle, then the third pair of angles are congruent.
Correct!
PTS: 1
DIF: Average
REF: Page 226
OBJ: 4-2.4 Applying the Third Angles Theorem
NAT: 12.3.3.f
STA: GE12.0
TOP: 4-2 Angle Relationships in Triangles
24
ID: A
47. ANS: B
CE = CD + DE
6x = (4x + 8) + 27
6x = 4x + 35
2x = 35
2x 35
=
2
2
35
x=
or 17.5
2
Segment Addition Postulate
Substitute 6x for CE and 4x + 8 for CD.
Simplify.
Subtract 4x from both sides.
Divide both sides by 2.
Simplify.
CE = 6x = 6 (17.5) = 105
Feedback
A
B
C
D
You found the value of x. Find the length of the specified segment.
Correct!
Check your equation. Make sure you are not subtracting instead of adding.
You found the length of a different segment.
PTS: 1
DIF: Average
REF: Page 15
OBJ: 1-2.3 Using the Segment Addition Postulate
NAT: 12.3.5.a
STA: GE1.0
TOP: 1-2 Measuring and Constructing Segments
KEY: segment addition postulate
48. ANS: B
A translation is a transformation where all the points of a figure are moved the same distance in the same
direction.
This transformation is not a translation because not all of the points have moved the same distance.
Feedback
A
B
Check where all the points have moved.
Correct!
PTS: 1
NAT: 12.3.2.c
DIF: Basic
STA: GE22.0
REF: Page 831
OBJ: 12-2.1 Identifying Translations
TOP: 12-2 Translations
25
ID: A
49. ANS: D
(x)° = (3x − 70)°
0 = 2x − 70
70 = 2x
35 = x
m∠ABC = 3x − 70
m∠ABC = 3(35) − 70 = 35°
Corresponding Angles Postulate
Subtract x from both sides.
Add 70 to both sides.
Divide both sides by 2.
Substitute 35 for x. Simplify.
Feedback
A
B
C
D
Use the Corresponding Angles Postulate.
First, set the measures of the corresponding angles equal to each other. Then, solve for
x and substitute in the expression (3x – 70).
If two parallel lines are cut by a transversal, then the pairs of corresponding angles are
congruent.
Correct!
PTS: 1
DIF: Average
REF: Page 155
OBJ: 3-2.1 Using the Corresponding Angles Postulate
NAT: 12.3.3.g
STA: GE7.0
TOP: 3-2 Angles Formed by Parallel Lines and Transversals
50. ANS: B
Proof:
Statements
Reasons
1. Given
1. m∠1 + m∠2 = 90°
2. m∠3 + m∠4 = 90°
2. Given
3. m∠1 + m∠2 = m∠3 + m∠4
3. Substitution Property
4. Given
4. m∠2 = m∠3
5. m∠1 + m∠2 = m∠2 + m∠4
5. Substitution Property
6. m∠1 = m∠4
6. Subtraction Property of Equality
Feedback
A
B
C
D
To get from Step 4 to Step 5, use substitution, not subtraction.
Correct!
To get from Step 4 to Step 5, use substitution, not addition.
Check the given information.
PTS: 1
DIF: Average
REF: Page 112
OBJ: 2-6.3 Writing a Two-Column Proof from a Plan
STA: GE2.0
TOP: 2-6 Geometric Proof
26
NAT: 12.3.5.a