Vertical Angles

Vertical angles are the angles that are opposite each other when two lines intersect. (Technically, these two lines need to be on the same plane)

In the picture on the left, $$ \angle $$ 1 and $$ \angle $$2 are vertical angles. Likewise, $$ \angle $$A and $$ \angle $$ B are vertical. Vertical angles are always congruent (equal).

Below is another picture of vertical angles. The blue pair and red pair of angles are congruent pairs of vertical angles.

Interactive Vertical Angles (Full Size Interactive Vertical Angles)

Click and the points below to see the rule for vertical angles in action.

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m$$ \angle $$ X  in the diagram on the left is 157° since its vertical angle is 157°.

Use the theorem that vertical angles are congruent to find the value of x in the problems below.

Practice Problems

Problem 1

What is the m $$ \angle $$ B on the left?

Angle B is 130°


Vertical Angle problems can also involve algebraic expressions. To find the value of x, set the two vertical angles equal then solve the equation:
  x+5=2x-3 x=8


Problem 2

What is the value of x?

Vertical Angles Example

x-2 =133 x = 135°

Problem 3

What is the value of x?

Vertical Angles Diagram

2x+5 = 105°
2x = 100
½(2x = 100) x = 50

Problem 4

Use the vertical angles theorem to solve for x

Vertical Angles Theorem

4x +7 = 131°
4x = 124
¼(4x = 124) x = 31

Problem 5

Find the value of x in the picture on the left

Solving Vertical Angles Problem
Problem 6

Use your knowledge of vertical angles to solve for x

Vertical Angles Problem
vertical angles equation
Problem 7

Solve for x

Vertical Angle Practice Problem
vertical angles equation
Problem 8

Use vertical angles to find the value of x

vertical angles equation
Problem 9

Use vertical angles to find the value of x

vertical angles equation
Problem 10

Use vertical angles to solve for x

vertical angles equation