Question
Question 1
(Multiple Choice Worth 1 Points)
(01.07 MC)
Use the figure and flowchart proof to answer the question:
Segments UV and WZ are parallel segments that intersect line ST at points Q and R, respectively
Points S, Q, R, and T all lie on the same line; Given. Arrows are drawn from this statement to the following three statements. Statement 1: The measure of angle SQT equals 180 degrees; Reason 1: Definition of a Straight Angle. Statement 2: The measure of angle SQV plus the measure of angle VQT equals the measure of angle SQT; Reason 2: Angle Addition Postulate. Statement 3: The measure of angle SQV plus the measure of angle VQT equals 180 degrees; Reason 3: Substitution Property of Equality. Lines UV and WZ are parallel; Given. An arrow is drawn from this statement to the following statements. Statement 4: The measure of angle VQT plus the measure of angle ZRS equals 180 degrees; Reason A. Statement 5: The measure of angle SQV plus the measure of angle VQT equals the measure of angle VQT plus the measure of angle ZRS; Reason B. An arrow also points from Statement 3 to Statement 5. An arrow from Statement 5 points to the following statements. Statement 6: The measure of angle SQV plus the measure of angle VQT minus the measure of angle VQT equals the measure of angle VQT plus the measure of angle ZRS minus the measure of angle VQT, the measure of angle SQV equals the measure of angle ZRS; Reason C. Statement 7: The measure of angle SQV is congruent to the measure of angle ZRS; Definition of Congruency.
Which property of equality accurately completes Reason B?
Addition Property of Equality
Division Property of Equality
Substitution Property of Equality
Subtraction Property of Equality
Question 2
(Multiple Choice Worth 1 Points)
(01.07 MC)
A pair of parallel lines is cut by a transversal, as shown in the figure:
A pair of parallel lines is cut by a transversal. The interior angle made on the left by the intersection of the upper parallel line and the transversal is labeled as x. The interior angle made on the right by the intersection of the lower parallel line and the transversal is labeled as y.
Which of the following best represents the relationship between angles x and y?
x = y
x = 180 degrees − y
y = 180 degrees − x
x = 2y
Question 3
(Multiple Choice Worth 1 Points)
(01.07 MC)
Use the figure to answer the question that follows:
segments UV and WZ are parallel with line ST intersecting both at points Q and R, respectively
The two-column proof below describes the statements and reasons for proving that corresponding angles are congruent:
Step Statements Reasons
1 segment UV is parallel to segment WZ Given
2 Points S, Q, R, and T all lie on the same line. Given
3 m∠SQT = 180° Definition of a Straight Angle
4 m∠SQV + m∠VQT = m∠SQT Angle Addition Postulate
5 Substitution Property of Equality
6 m∠VQT + m∠ZRS = 180° Same-Side Interior Angles Theorem
7 m∠SQV + m∠VQT = m∠VQT + m∠ZRS Substitution Property of Equality
8 m∠SQV + m∠VQT − m∠VQT = m∠VQT + m∠ZRS − m∠VQT
m∠SQV = m∠ZRS Subtraction Property of Equality
∠SQV ≅ ∠ZRS Definition of Congruency
What is the missing statement for step 5?
m
∠
SQV =
m
∠
SQT
m
∠
SQV +
m
∠
VQT = 180°
m
∠
SQV +
m
∠
SQT =
m
∠
VQT
m
∠
SQV +
m
∠
SQT = 180°
Question 4
(Multiple Choice Worth 1 Points)
(01.07 MC)
The following is an incomplete paragraph proving that ∠WRS ≅ ∠VQT, given the information in the figure where segment UV is parallel to segment WZ.:
Segments UV and WZ are parallel segments that intersect with line ST at points Q and R, respectively.
According to the given information, segment UV is parallel to segment WZ, while angles SQU and VQT are vertical angles. ________ by the Vertical Angles Theorem. Because angles SQU and WRS are corresponding angles, they are congruent according to the Corresponding Angles Theorem. Finally, angle VQT is congruent to angle WRS by the Transitive Property of Equality.
Which phrase accurately completes the proof?
∠
SQU ≅
∠
VQT
∠
SQU ≅
∠
WRS
∠
WRS ≅
∠
VQT
∠
WRS ≅
∠
ZRT
Question 5
(Multiple Choice Worth 1 Points)
(01.07 MC)
A pair of parallel lines is cut by a transversal.
A pair of parallel lines is shown cut by a transversal. Angle A is located in the upper left exterior next to the transversal, and angle B is located in the bottom right exterior corner of the transversal.
If m∠A = (5x − 4)° and m∠B = (8x − 28)°, what is the value of x?
8
9.4
16.3
36
Question 6
(Multiple Choice Worth 1 Points)
(01.07 MC)
Lines AC and BD intersect at point O.
Lines AC and BD intersect at point O.
If m∠AOD = (5x − 4)° and m∠BOC = (4x + 5)°, what is m∠BOC?
9°
10°
41°
82°
Question 7
(Multiple Choice Worth 1 Points)
(01.07 MC)
Kelly and Daniel wrote the following proofs to prove that vertical angles are congruent. Who is correct?
Line segment NT intersects line segment MR, forming four angles. Angles 1 and 3 are vertical angles. Angles 2 and 4 are vertical angles.
Kelly's Proof
Statement Justification
∠2 = ∠4 Vertical angles are congruent.
∠1 = ∠3 Vertical angles are congruent.
Vertical angles are congruent. Vertical Angle Theorem
Daniel's Proof
Statement Justification
∠1 + ∠2 = 180° Definition of Supplementary Angles
∠1 + ∠4 = 180° Definition of Supplementary Angles
∠1 + ∠2 = ∠1 + ∠4 Substitution Property of Equality
∠2 = ∠4 Subtraction Property of Equality
Both Kelly and Daniel are correct.
Neither Kelly nor Daniel is correct.
Kelly is correct, but Daniel is not.
Daniel is correct, but Kelly is not.
Question 8
(Multiple Choice Worth 1 Points)
(01.07 MC)
Lines BC and ED are parallel. They are intersected by transversal AE, in which point B lies between points and E. They are also intersected by transversal EC. Angle ABC measures 70 degrees. Angle CED measures 30 degrees.
Given: line BC is parallel to line ED
m∠ABC = 70°
m∠CED = 30°
Prove: m∠BEC = 40°
Statement Justification
line BC is parallel to line ED Given
m∠ABC = 70° Given
m∠CED = 30° Given
m∠ABC = m∠BED Corresponding Angles Theorem
m∠BEC + m∠CED = m∠BED Angle Addition Postulate
m∠BEC = 40° Subtraction Property of Equality
Which of the following accurately completes the missing statement and justification of the two-column proof?
m
∠
BEC + 30° = 70°; Substitution Property of Equality
m
∠
BEC + 30° = 70°; Addition Property of Equality
m
∠
BEC + 40° = 70°; Substitution Property of Equality
m
∠
BEC + 40° = 70°; Addition Property of Equality
Question 9
(Multiple Choice Worth 1 Points)
(01.07 MC)
The figure shown has two parallel lines cut by a transversal:
A pair of parallel lines is shown, crossed by a transversal. Angles are identified as 1, 2, 3, 4, 5, 6, 7, and 8. Angles 1-4 are on the top line clockwise from upper left. Angles 5-8 are on the lower line clockwise from upper left.
Which angle is a corresponding angle to ∠6?
∠
2
∠
3
∠
7
∠
8
Question 10
(Multiple Choice Worth 1 Points)
(01.07 MC)
Two friends are sharing one half of a pizza that is divided into three pieces, as shown in the figure.
Circle O has diameter A D and radii B O and C O. Angle A O B is adjacent to angle B O C. Angle B O C is adjacent to angle C O D.
If m∠AOB = (6x − 5)° and m∠BOD = (9x + 17)°, what is the value of x?
117.8
62.2
11.2
7.3
(Multiple Choice Worth 1 Points)
(01.07 MC)
Use the figure and flowchart proof to answer the question:
Segments UV and WZ are parallel segments that intersect line ST at points Q and R, respectively
Points S, Q, R, and T all lie on the same line; Given. Arrows are drawn from this statement to the following three statements. Statement 1: The measure of angle SQT equals 180 degrees; Reason 1: Definition of a Straight Angle. Statement 2: The measure of angle SQV plus the measure of angle VQT equals the measure of angle SQT; Reason 2: Angle Addition Postulate. Statement 3: The measure of angle SQV plus the measure of angle VQT equals 180 degrees; Reason 3: Substitution Property of Equality. Lines UV and WZ are parallel; Given. An arrow is drawn from this statement to the following statements. Statement 4: The measure of angle VQT plus the measure of angle ZRS equals 180 degrees; Reason A. Statement 5: The measure of angle SQV plus the measure of angle VQT equals the measure of angle VQT plus the measure of angle ZRS; Reason B. An arrow also points from Statement 3 to Statement 5. An arrow from Statement 5 points to the following statements. Statement 6: The measure of angle SQV plus the measure of angle VQT minus the measure of angle VQT equals the measure of angle VQT plus the measure of angle ZRS minus the measure of angle VQT, the measure of angle SQV equals the measure of angle ZRS; Reason C. Statement 7: The measure of angle SQV is congruent to the measure of angle ZRS; Definition of Congruency.
Which property of equality accurately completes Reason B?
Addition Property of Equality
Division Property of Equality
Substitution Property of Equality
Subtraction Property of Equality
Question 2
(Multiple Choice Worth 1 Points)
(01.07 MC)
A pair of parallel lines is cut by a transversal, as shown in the figure:
A pair of parallel lines is cut by a transversal. The interior angle made on the left by the intersection of the upper parallel line and the transversal is labeled as x. The interior angle made on the right by the intersection of the lower parallel line and the transversal is labeled as y.
Which of the following best represents the relationship between angles x and y?
x = y
x = 180 degrees − y
y = 180 degrees − x
x = 2y
Question 3
(Multiple Choice Worth 1 Points)
(01.07 MC)
Use the figure to answer the question that follows:
segments UV and WZ are parallel with line ST intersecting both at points Q and R, respectively
The two-column proof below describes the statements and reasons for proving that corresponding angles are congruent:
Step Statements Reasons
1 segment UV is parallel to segment WZ Given
2 Points S, Q, R, and T all lie on the same line. Given
3 m∠SQT = 180° Definition of a Straight Angle
4 m∠SQV + m∠VQT = m∠SQT Angle Addition Postulate
5 Substitution Property of Equality
6 m∠VQT + m∠ZRS = 180° Same-Side Interior Angles Theorem
7 m∠SQV + m∠VQT = m∠VQT + m∠ZRS Substitution Property of Equality
8 m∠SQV + m∠VQT − m∠VQT = m∠VQT + m∠ZRS − m∠VQT
m∠SQV = m∠ZRS Subtraction Property of Equality
∠SQV ≅ ∠ZRS Definition of Congruency
What is the missing statement for step 5?
m
∠
SQV =
m
∠
SQT
m
∠
SQV +
m
∠
VQT = 180°
m
∠
SQV +
m
∠
SQT =
m
∠
VQT
m
∠
SQV +
m
∠
SQT = 180°
Question 4
(Multiple Choice Worth 1 Points)
(01.07 MC)
The following is an incomplete paragraph proving that ∠WRS ≅ ∠VQT, given the information in the figure where segment UV is parallel to segment WZ.:
Segments UV and WZ are parallel segments that intersect with line ST at points Q and R, respectively.
According to the given information, segment UV is parallel to segment WZ, while angles SQU and VQT are vertical angles. ________ by the Vertical Angles Theorem. Because angles SQU and WRS are corresponding angles, they are congruent according to the Corresponding Angles Theorem. Finally, angle VQT is congruent to angle WRS by the Transitive Property of Equality.
Which phrase accurately completes the proof?
∠
SQU ≅
∠
VQT
∠
SQU ≅
∠
WRS
∠
WRS ≅
∠
VQT
∠
WRS ≅
∠
ZRT
Question 5
(Multiple Choice Worth 1 Points)
(01.07 MC)
A pair of parallel lines is cut by a transversal.
A pair of parallel lines is shown cut by a transversal. Angle A is located in the upper left exterior next to the transversal, and angle B is located in the bottom right exterior corner of the transversal.
If m∠A = (5x − 4)° and m∠B = (8x − 28)°, what is the value of x?
8
9.4
16.3
36
Question 6
(Multiple Choice Worth 1 Points)
(01.07 MC)
Lines AC and BD intersect at point O.
Lines AC and BD intersect at point O.
If m∠AOD = (5x − 4)° and m∠BOC = (4x + 5)°, what is m∠BOC?
9°
10°
41°
82°
Question 7
(Multiple Choice Worth 1 Points)
(01.07 MC)
Kelly and Daniel wrote the following proofs to prove that vertical angles are congruent. Who is correct?
Line segment NT intersects line segment MR, forming four angles. Angles 1 and 3 are vertical angles. Angles 2 and 4 are vertical angles.
Kelly's Proof
Statement Justification
∠2 = ∠4 Vertical angles are congruent.
∠1 = ∠3 Vertical angles are congruent.
Vertical angles are congruent. Vertical Angle Theorem
Daniel's Proof
Statement Justification
∠1 + ∠2 = 180° Definition of Supplementary Angles
∠1 + ∠4 = 180° Definition of Supplementary Angles
∠1 + ∠2 = ∠1 + ∠4 Substitution Property of Equality
∠2 = ∠4 Subtraction Property of Equality
Both Kelly and Daniel are correct.
Neither Kelly nor Daniel is correct.
Kelly is correct, but Daniel is not.
Daniel is correct, but Kelly is not.
Question 8
(Multiple Choice Worth 1 Points)
(01.07 MC)
Lines BC and ED are parallel. They are intersected by transversal AE, in which point B lies between points and E. They are also intersected by transversal EC. Angle ABC measures 70 degrees. Angle CED measures 30 degrees.
Given: line BC is parallel to line ED
m∠ABC = 70°
m∠CED = 30°
Prove: m∠BEC = 40°
Statement Justification
line BC is parallel to line ED Given
m∠ABC = 70° Given
m∠CED = 30° Given
m∠ABC = m∠BED Corresponding Angles Theorem
m∠BEC + m∠CED = m∠BED Angle Addition Postulate
m∠BEC = 40° Subtraction Property of Equality
Which of the following accurately completes the missing statement and justification of the two-column proof?
m
∠
BEC + 30° = 70°; Substitution Property of Equality
m
∠
BEC + 30° = 70°; Addition Property of Equality
m
∠
BEC + 40° = 70°; Substitution Property of Equality
m
∠
BEC + 40° = 70°; Addition Property of Equality
Question 9
(Multiple Choice Worth 1 Points)
(01.07 MC)
The figure shown has two parallel lines cut by a transversal:
A pair of parallel lines is shown, crossed by a transversal. Angles are identified as 1, 2, 3, 4, 5, 6, 7, and 8. Angles 1-4 are on the top line clockwise from upper left. Angles 5-8 are on the lower line clockwise from upper left.
Which angle is a corresponding angle to ∠6?
∠
2
∠
3
∠
7
∠
8
Question 10
(Multiple Choice Worth 1 Points)
(01.07 MC)
Two friends are sharing one half of a pizza that is divided into three pieces, as shown in the figure.
Circle O has diameter A D and radii B O and C O. Angle A O B is adjacent to angle B O C. Angle B O C is adjacent to angle C O D.
If m∠AOB = (6x − 5)° and m∠BOD = (9x + 17)°, what is the value of x?
117.8
62.2
11.2
7.3
Answers
GPT-4o mini
Here are the answers to the multiple-choice questions:
**Question 1:**
Which property of equality accurately completes Reason B?
- **Answer:** Substitution Property of Equality
**Question 2:**
Which of the following best represents the relationship between angles x and y?
- **Answer:** x = y
**Question 3:**
What is the missing statement for step 5?
- **Answer:** m∠SQV = m∠SQT
**Question 4:**
Which phrase accurately completes the proof?
- **Answer:** ∠SQU ≅ ∠VQT
**Question 5:**
If m∠A = (5x − 4)° and m∠B = (8x − 28)°, what is the value of x?
- **Answer:** 8
**Question 6:**
What is m∠BOC?
- **Answer:** 41°
**Question 7:**
Who is correct regarding the proofs of vertical angles?
- **Answer:** Both Kelly and Daniel are correct.
**Question 8:**
Which statement and justification accurately complete the proof?
- **Answer:** m∠BEC + 30° = 70°; Addition Property of Equality
**Question 9:**
Which angle is a corresponding angle to ∠6?
- **Answer:** ∠8
**Question 10:**
If m∠AOB = (6x − 5)° and m∠BOD = (9x + 17)°, what is the value of x?
- **Answer:** 11.2
If you have any more questions or need explanations for any specific answers, please let me know!
**Question 1:**
Which property of equality accurately completes Reason B?
- **Answer:** Substitution Property of Equality
**Question 2:**
Which of the following best represents the relationship between angles x and y?
- **Answer:** x = y
**Question 3:**
What is the missing statement for step 5?
- **Answer:** m∠SQV = m∠SQT
**Question 4:**
Which phrase accurately completes the proof?
- **Answer:** ∠SQU ≅ ∠VQT
**Question 5:**
If m∠A = (5x − 4)° and m∠B = (8x − 28)°, what is the value of x?
- **Answer:** 8
**Question 6:**
What is m∠BOC?
- **Answer:** 41°
**Question 7:**
Who is correct regarding the proofs of vertical angles?
- **Answer:** Both Kelly and Daniel are correct.
**Question 8:**
Which statement and justification accurately complete the proof?
- **Answer:** m∠BEC + 30° = 70°; Addition Property of Equality
**Question 9:**
Which angle is a corresponding angle to ∠6?
- **Answer:** ∠8
**Question 10:**
If m∠AOB = (6x − 5)° and m∠BOD = (9x + 17)°, what is the value of x?
- **Answer:** 11.2
If you have any more questions or need explanations for any specific answers, please let me know!