Vertical Angles in Diagrams � Apply Angle Relationships (7.G.B.5)

1) ∠A and ∠C are vertical angles. If ∠A = 95°, what is ∠C?

85°

95°

105°

110°

Vertical angles are always congruent.

Step 1: Set them equal → ∠A = ∠C.

Step 2: Substitute → ∠C = 95°.

Final Answer: ∠C = 95°

Explanation

Vertical angles are always congruent.

Step 1: Set them equal → ∠A = ∠C.

Step 2: Substitute → ∠C = 95°.

Final Answer: ∠C = 95°

2) Two intersecting lines form four angles labeled ∠1, ∠2, ∠3, and ∠4. If ∠2 = 110°, what is ∠4?

70°

90°

110°

120°

Vertical angles are equal in measure.

Step 1: ∠2 and ∠4 are opposite each other.

Step 2: So ∠4 = 110°.

Final Answer: ∠4 = 110°

Explanation

Vertical angles are equal in measure.

Step 1: ∠2 and ∠4 are opposite each other.

Step 2: So ∠4 = 110°.

Final Answer: ∠4 = 110°

3) At an intersection, ∠M = 45°. Which angle is its vertical angle?

The angle next to ∠M

The angle across from ∠M

Both adjacent angles

None of the above

Vertical angles are the pair across from each other, not beside.

Explanation

Vertical angles are the pair across from each other, not beside.

4) In the diagram, ∠1 = 80°. What is the measure of the vertical angle to ∠1?

80°

100°

120°

180°

Vertical angles are congruent.

So, the vertical angle to ∠1 = 80°.

Explanation

Vertical angles are congruent.

So, the vertical angle to ∠1 = 80°.

5) At an intersection, ∠A and ∠B are vertical. ∠B is complementary to ∠C. If m∠A = 4x + 6 and m∠C = x + 24, what is m∠A?

36°

54°

66°

72°

Step 1: ∠A = ∠B (vertical).

Step 2: ∠B + ∠C = 90° (complementary).

Step 3: (4x + 6) + (x + 24) = 90 → 5x + 30 = 90 → x = 12.

Step 4: ∠A = 4(12) + 6 = 54°.°.

Explanation

Step 1: ∠A = ∠B (vertical).

Step 2: ∠B + ∠C = 90° (complementary).

Step 3: (4x + 6) + (x + 24) = 90 → 5x + 30 = 90 → x = 12.

Step 4: ∠A = 4(12) + 6 = 54°.°.

6) Two streets intersect, forming four angles. One angle measures 62°. What is the measure of the vertical angle across from it?

28°

62°

118°

180°

Vertical angles are equal.

So, the opposite angle = 62°.

Explanation

Vertical angles are equal.

So, the opposite angle = 62°.

7) If ∠1 = 75°, and ∠3 is its vertical angle, what is ∠3?

75°

90°

105°

115°

Vertical angles are always equal → ∠3 = 75°.

Explanation

Vertical angles are always equal → ∠3 = 75°.

8) Two lines intersect. One angle is 125°. What is its vertical angle?

55°

65°

115°

125°

Vertical angles are equal. So, both measure 125°.

Explanation

Vertical angles are equal. So, both measure 125°.

9) At an intersection, ∠x = 90°. What is the vertical angle to ∠x?

45°

60°

90°

180°

When lines intersect forming right angles, all four angles = 90°.

Explanation

When lines intersect forming right angles, all four angles = 90°.

10) If ∠y = 137°, what is the measure of its vertical angle?

43°

137°

143°

180°

Vertical angles are equal → vertical = 137°.

Explanation

Vertical angles are equal → vertical = 137°.

11) Two vertical angles are marked ∠A = 120° and ∠B. What is ∠B?

30°

60°

120°

180°

Vertical angles are equal.

So, ∠B = 120°.

Explanation

Vertical angles are equal.

So, ∠B = 120°.

12) Two lines intersect at point OOO. Angles are labeled ∠1, ∠2, ∠3, ∠4 in order around O. Which option correctly lists all vertical pairs and all linear pairs?

Vertical: (1,2), (3,4); Linear: (1,3), (2,4)

Vertical: (1,3), (2,4); Linear: (1,2), (2,3), (3,4), (4,1)

Vertical: (1,4), (2,3); Linear: (1,3), (2,4)

Vertical: (1,3), (2,4); Linear: (1,4), (2,4), (1,3), (2,3)

Vertical pairs are opposite: (1,3) and (2,4). Linear pairs are adjacent: (1,2), (2,3), (3,4), (4,1).

So, the vertical pairs are (1,3) and (2,4), and the linear pairs are (1,2), (2,3), (3,4), (4,1).

Explanation

Vertical pairs are opposite: (1,3) and (2,4). Linear pairs are adjacent: (1,2), (2,3), (3,4), (4,1).

So, the vertical pairs are (1,3) and (2,4), and the linear pairs are (1,2), (2,3), (3,4), (4,1).

13) If one angle is 15°, what is its vertical angle?

15°

30°

45°

165°

Vertical angles are equal, so both are 15°.

Explanation

Vertical angles are equal, so both are 15°.

14) Which of the following statements about vertical angles is true?

They are adjacent and congruent.

They are opposite and congruent.

They always add to 90°.

They always add to 180°.

Vertical angles sit opposite each other and are congruent.

So, they are opposite and congruent.

Explanation

Vertical angles sit opposite each other and are congruent.

So, they are opposite and congruent.

15) Two lines intersect. One angle is 118°. What are the measures of the other three angles?

62°, 62°, 118°

62°, 118°, 118°

62°, 62°, 62°

118°, 118°, 118°

Step 1: Vertical opposite = 118°.

Step 2: Adjacent = 180 − 118 = 62°.

Step 3: Opposite to 62° = 62°.

Explanation

Step 1: Vertical opposite = 118°.

Step 2: Adjacent = 180 − 118 = 62°.

Step 3: Opposite to 62° = 62°.

16) In the diagram, ∠1 = (2x + 10)° and ∠3 = (3x – 20)°. If ∠1 and ∠3 are vertical, what is x?

15

20

25

30

Step 1: Set equal → 2x + 10 = 3x – 20.

Step 2: Simplify → x = 30.

Step 3: Substitute → 2(30) + 10 = 70°.

Explanation

Step 1: Set equal → 2x + 10 = 3x – 20.

Step 2: Simplify → x = 30.

Step 3: Substitute → 2(30) + 10 = 70°.

17) Two lines intersect. ∠1 = (4x – 15)°, ∠2 = (2x + 25)°. If they are vertical angles, what is ∠1?

35°

45°

55°

65°

Vertical: 4x − 15 = 2x + 25 → 2x = 40 → x = 20.

Then ∠1 = 4(20) − 15 = 65.

The measure of ∠1 is 65°.

Explanation

Vertical: 4x − 15 = 2x + 25 → 2x = 40 → x = 20.

Then ∠1 = 4(20) − 15 = 65.

The measure of ∠1 is 65°.

18) In the diagram, ∠1 = 70°. Which angle is vertical to ∠1?

∠2

∠3

∠4

∠5

Vertical angles are opposite each other when two lines cross.

So, if ∠1 = 70°, the angle directly across (∠4) is also 70°.

Explanation

Vertical angles are opposite each other when two lines cross.

So, if ∠1 = 70°, the angle directly across (∠4) is also 70°.

19) Two lines intersect. One angle measures 2x+14 and its vertical angle measures 5x−31. What is the measure of an angle adjacent to them?

116°

124°

136°

144°

Vertical angles are equal. So,

Step 1: 2x + 14 = 5x − 31

Step 2: 45 = 3x → x = 15.

Step 3: Substitute → 2(15) + 14 = 44°.

Adjacent angles form a linear pair → 180 − 44 = 136°.

Explanation

Vertical angles are equal. So,

Step 1: 2x + 14 = 5x − 31

Step 2: 45 = 3x → x = 15.

Step 3: Substitute → 2(15) + 14 = 44°.

Adjacent angles form a linear pair → 180 − 44 = 136°.

20) In the diagram, ∠A = 100°. Which angle is congruent to ∠A?

Adjacent angle

Vertical angle

Complementary angle

Supplementary angle

Vertical angles are congruent → the one across from ∠A equals 100°.

Explanation

Vertical angles are congruent → the one across from ∠A equals 100°.

Applying Vertical Angle Relationships in Diagrams

1) ∠A and ∠C are vertical angles. If ∠A = 95°, what is ∠C?

2) Two intersecting lines form four angles labeled ∠1, ∠2, ∠3, and ∠4. If ∠2 = 110°, what is ∠4?

3) At an intersection, ∠M = 45°. Which angle is its vertical angle?

4) In the diagram, ∠1 = 80°. What is the measure of the vertical angle to ∠1?

5) At an intersection, ∠A and ∠B are vertical. ∠B is complementary to ∠C. If m∠A = 4x + 6 and m∠C = x + 24, what is m∠A?

6) Two streets intersect, forming four angles. One angle measures 62°. What is the measure of the vertical angle across from it?

7) If ∠1 = 75°, and ∠3 is its vertical angle, what is ∠3?

8) Two lines intersect. One angle is 125°. What is its vertical angle?

9) At an intersection, ∠x = 90°. What is the vertical angle to ∠x?

10) If ∠y = 137°, what is the measure of its vertical angle?

11) Two vertical angles are marked ∠A = 120° and ∠B. What is ∠B?

12) Two lines intersect at point OOO. Angles are labeled ∠1, ∠2, ∠3, ∠4 in order around O. Which option correctly lists all vertical pairs and all linear pairs?

13) If one angle is 15°, what is its vertical angle?

14) Which of the following statements about vertical angles is true?

15) Two lines intersect. One angle is 118°. What are the measures of the other three angles?

16) In the diagram, ∠1 = (2x + 10)° and ∠3 = (3x – 20)°. If ∠1 and ∠3 are vertical, what is x?

17) Two lines intersect. ∠1 = (4x – 15)°, ∠2 = (2x + 25)°. If they are vertical angles, what is ∠1?

18) In the diagram, ∠1 = 70°. Which angle is vertical to ∠1?

19) Two lines intersect. One angle measures 2x+14 and its vertical angle measures 5x−31. What is the measure of an angle adjacent to them?

20) In the diagram, ∠A = 100°. Which angle is congruent to ∠A?