Solving Equations with Vertical Angles

8th Grade

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Quizzes Created: 8157 | Total Attempts: 9,569,759

| Questions: 20 | Updated: Jan 16, 2026

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1) If ∠x = 3y + 5 and ∠y = 50, and the two angles are vertical, what is the value of y?

Step 1: Vertical angles are equal, so set 3y + 5 = 50.

Step 2: Subtract 5 → 3y = 45.

Step 3: Divide by 3 → y = 15.

Final Answer: y = 15

Explanation

Step 1: Vertical angles are equal, so set 3y + 5 = 50.

Step 2: Subtract 5 → 3y = 45.

Step 3: Divide by 3 → y = 15.

Final Answer: y = 15

Submit

About This Quiz

Solving Equations With Vertical Angles - Quiz

Can you combine algebra and geometry? In this quiz, you’ll solve equations based on the fact that vertical angles are always equal. You’ll begin with straightforward setups and then tackle multi-step problems involving angles that form straight lines. Using what you know about vertical and supplementary angles, you’ll find missing... see morevalues while seeing how algebra and geometry work hand in hand.
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2) Two vertical angles are given as (2x + 10)° and (4x – 30)°. Find the measure of each angle.

40°

50°

55°

60°

Step 1: Vertical angles are equal → 2x + 10 = 4x – 30.

Step 2: Simplify → 40 = 2x → x = 20.

Step 3: Substitute → 2(20) + 10 = 50.

Explanation

Step 1: Vertical angles are equal → 2x + 10 = 4x – 30.

Step 2: Simplify → 40 = 2x → x = 20.

Step 3: Substitute → 2(20) + 10 = 50.

Submit

3) If ∠A = 5x + 15 and ∠B = 95, and ∠A and ∠B are vertical angles, what is x?

Step 1: Set equal → 5x + 15 = 95.

Step 2: Subtract 15 → 5x = 80.

Step 3: Divide by 5 → x = 16.

Explanation

Step 1: Set equal → 5x + 15 = 95.

Step 2: Subtract 15 → 5x = 80.

Step 3: Divide by 5 → x = 16.

Submit

4) ∠1 = (6y – 20)°, ∠2 = (4y + 10)°. If ∠1 and ∠2 are vertical, then what is y?

Vertical angles are equal. So,

Step 1: Set equal → 6y – 20 = 4y + 10.

Step 2: Simplify → 2y = 30.

Step 3: Divide → y = 15.

Explanation

Vertical angles are equal. So,

Step 1: Set equal → 6y – 20 = 4y + 10.

Step 2: Simplify → 2y = 30.

Step 3: Divide → y = 15.

Submit

5) Two vertical angles are represented as (x + 30)° and (2x – 60)°. What is the value of x?

Vertical angles are equal. So,

Step 1: Set equal → x + 30 = 2x – 60.

Step 2: Simplify → 90 = x.

Step 3: Substitute to check → (90 + 30) = 120°, both match.

Explanation

Vertical angles are equal. So,

Step 1: Set equal → x + 30 = 2x – 60.

Step 2: Simplify → 90 = x.

Step 3: Substitute to check → (90 + 30) = 120°, both match.

Submit

6) If one vertical angle is (7x – 25)° and the other is (5x + 15)°, find the value of x.

Vertical angles are equal. So,

Step 1: Set equal → 7x – 25 = 5x + 15.

Step 2: Simplify → 2x = 40.

Step 3: Divide → x = 20.

Explanation

Vertical angles are equal. So,

Step 1: Set equal → 7x – 25 = 5x + 15.

Step 2: Simplify → 2x = 40.

Step 3: Divide → x = 20.

Submit

7) ∠M = (3a + 5)°, ∠N = (2a + 25)°. If ∠M and ∠N are vertical, then what is a?

Vertical angles are equal. So,

Step 1: Set equal → 3a + 5 = 2a + 25.

Step 2: Simplify → a = 20.

Step 3: Substitute → ∠M = 3(20) + 5 = 65° and ∠N = 2(20) + 25 = 65°.

Explanation

Vertical angles are equal. So,

Step 1: Set equal → 3a + 5 = 2a + 25.

Step 2: Simplify → a = 20.

Step 3: Substitute → ∠M = 3(20) + 5 = 65° and ∠N = 2(20) + 25 = 65°.

Submit

8) If vertical angles are expressed as (x – 10)° and (2x – 40)°, what is x?

Vertical angles are equal. So,

Step 1: Set equal → x + 30 = 2x – 60.

Step 2: Simplify → x = 90.

Step 3: Substitute → (90 + 30) = 120°, (2×90 – 60) = 120°.

Final Answer: x = 90, and each angle = 120°.

Explanation

Submit

9) Two vertical angles are (4b – 5)° and (3b + 10)°. Find the measure of each angle.

35°

40°

45°

55°

Vertical angles are equal. So,

Step 1: Set equal → 4b – 5 = 3b + 10.

Step 2: Simplify → b = 15.

Step 3: Substitute → (4×15 – 5) = 55° and (3×15 + 10) = 55°.

Final Answer: 15, and each angle = 55°.

Explanation

Submit

10) ∠1 and ∠2 are vertical. ∠2 and ∠3 form a linear pair. If ∠1 = 3x + 6 and ∠3 = x + 66, find the measures of ∠1, ∠2, ∠3 (in that order).

87°, 87°, 93°

93°, 93°, 87

87°, 93°, 93°

90°, 90°, 90°

Vertical angles are equal. So,

Step 1: ∠1 = ∠2 = 3x + 6, and ∠2 + ∠3 = 180.

Step 2: Substitute → (3x + 6) + (x + 66) = 180 → 4x + 72 = 180 → x = 27.

Step 3: ∠1 = 3(27) + 6 = 87°, ∠3 = 93°.

Final Answer: ∠1 = 87°, ∠2 = 87°, ∠3 = 93°

Explanation

Submit

11) Vertical angles are written as (12 + 2x)° and (3x – 6)°. What is the measure of the angles?

18°

30°

48°

55°

Vertical angles are equal. So,

Step 1: Set equal → 12 + 2x = 3x – 6.

Step 2: Simplify → x = 18.

Step 3: Substitute → 12 + 2(18) = 48°.

Final Answer: Each angle = 48°

Explanation

Vertical angles are equal. So,

Step 1: Set equal → 12 + 2x = 3x – 6.

Step 2: Simplify → x = 18.

Step 3: Substitute → 12 + 2(18) = 48°.

Final Answer: Each angle = 48°

Submit

12) ∠P = (6k – 12)°, ∠Q = (4k + 8)°. If P and Q are vertical, find k.

Vertical angles are equal. So,

Step 1: Set equal → 6k – 12 = 4k + 8.

Step 2: Simplify → 2k = 20.

Step 3: Divide → k = 10.

Final Answer: k = 10

Explanation

Vertical angles are equal. So,

Step 1: Set equal → 6k – 12 = 4k + 8.

Step 2: Simplify → 2k = 20.

Step 3: Divide → k = 10.

Final Answer: k = 10

Submit

13) If ∠x = (5m + 7)° and ∠y = (6m – 3)°, find the angle measure if x and y are vertical.

47°

52°

57°

62°

Vertical angles are equal. So,

Step 1: Set equal → 5m + 7 = 6m – 3.

Step 2: Simplify → m = 10.

Step 3: Substitute → 5(10) + 7 = 57°.

Final Answer: Each angle = 57°

Explanation

Vertical angles are equal. So,

Step 1: Set equal → 5m + 7 = 6m – 3.

Step 2: Simplify → m = 10.

Step 3: Substitute → 5(10) + 7 = 57°.

Final Answer: Each angle = 57°

Submit

14) Two vertical angles are expressed as (2x + 25)° and (75 + x)°. Find x.

Vertical angles are equal. So,

Step 1: Set equal → 2x + 25 = 75 + x.

Step 2: Simplify → x = 50.

Step 3: Substitute → (2×50 + 25) = 125°, (75 + 50) = 125°.

Final Answer: x = 50, and each angle = 125°.

Explanation

Submit

15) If ∠A = (x + 12)° and ∠B = (2x – 18)°, and ∠A = ∠B, find the measure of each.

38°

42°

56°

60°

Vertical angles are equal. So,

Step 1: Set equal → x + 12 = 2x – 18.

Step 2: Simplify → x = 30.

Step 3: Substitute → (30 + 12) = 42°, (2×30 – 18) = 42°.

Final Answer: x = 30, and each angle = 42°.

Explanation

Submit

16) ∠1 and ∠2 are vertical. ∠2 and ∠3 form a linear pair. If ∠1 = 4y + 8 and ∠3 = 2y + 52, find the measures of ∠1, ∠2, ∠3 (in that order).

92°, 92°, 88°

88°, 92°, 88°

90°, 90°, 90°

88°, 88°, 92°

Vertical angles are equal. So,

Step 1: ∠1 = ∠2 = 4y + 8 and ∠2 + ∠3 = 180.

Step 2: Substitute → (4y + 8) + (2y + 52) = 180 → 6y + 60 = 180 → y = 20.

Step 3: Substitute → ∠1 = 4(20) + 8 = 88°, ∠3 = 2(20) + 52 = 92°.

Final Answer: ∠1 = 88°, ∠2 = 88°, ∠3 = 92°.

Explanation

Submit

17) Two vertical angles are (x – 3)° and (2x – 23)°. Find the measure of each.

10°

15°

17°

20°

Vertical angles are equal. So,

Step 1: Set equal → x – 3 = 2x – 23.

Step 2: Simplify → x = 20.

Step 3: Substitute → (20 – 3) = 17°, (2×20 – 23) = 17°.

Final Answer: x = 20, and each angle = 17°.

Explanation

Submit

18) ∠1 and ∠2 are vertical angles. ∠1 = (11n – 15)°, ∠2 = (7n + 21)°. Solve for n.

Vertical angles are equal. So,

Step 1: Set equal → 11n – 15 = 7n + 21.

Step 2: Simplify → 4n = 36.

Step 3: Divide → n = 9, and substitute → 11(9) – 15 = 84°.

Final Answer: n = 9, and each angle = 84°.

Explanation

Submit

19) If vertical angles ∠R = (4x + 12)° and ∠S = (6x – 18)°, find the value of x.

Vertical angles are equal. So,

Step 1: Set equal → 4x + 12 = 6x – 18.

Step 2: Simplify → 2x = 30.

Step 3: Divide → x = 15, and substitute → 4(15) + 12 = 72°.

Final Answer: x = 15, and each angle = 72°.

Explanation