Vertical Angles Theorem – Examples and Practice Problems

The vertical angle theorem tells us that the pairs of vertical angles formed by the intersection of two lines have the same size. In this article, we will look at a summary of the vertical angles theorem.

Then, we will apply this theorem by solving various practice problems.

GEOMETRY
example 3 of vertical angles theorem

Relevant for

Learning about the vertical angles theorem with examples.

See examples

GEOMETRY
example 3 of vertical angles theorem

Relevant for

Learning about the vertical angles theorem with examples.

See examples

Summary of the vertical angles theorem

Vertical angles are the angles formed by the intersection of two lines. For example, in the diagram below, we have two pairs of vertical angles.

diagram of opposite vertical angles

Angles a and b and angles c and d are pairs of vertical angles.

The vertical angles theorem tells us that pairs of vertical angles have the same size. Therefore, in the diagram shown above, we know that angles a and b and angles c and d are equal.

You can look at the proof of this theorem in this article.


Vertical angles theorem – Examples with answers

The following examples are solved using the vertical angles theorem. Try to solve the exercises yourself before looking at the solution.

EXAMPLE 1

Determine the size of each of the angles in the diagram below.

example 1 of vertical angles theorem

Angles ∠62° and ∠b are a pair of vertical angles. Thus, we have:

∠b = 62°

The value of ∠a can be found considering that it is a supplementary angle to the angle of 62°. That is, these angles add up to 180° and we have:

62° + ∠a = 180°

∠a = 118°

Finally, we can see that angles ∠a and ∠c are another pair of vertical angles, so we have:

∠c = 118°

EXAMPLE 2

What is the value of Y in the diagram below?

opposite angles by the vertex example 2

Angles ∠100° and ∠X are part of a straight line, so they are supplementary. Therefore, we have:

100° + ∠X = 180°

∠X = 80°

Since the angles (Y + 30)° and X are vertical angles, we can form the following equation and solve for Y:

Y + 30 = 80

Y = 50°

EXAMPLE 3

What is the value of Z in the diagram below?

example 3 of vertical angles theorem

This problem is similar to the previous one. Therefore, we know that angles ∠X and ∠110° form a straight line, so they are supplementary. Then, we have:

110° + ∠X = 180°

∠X = 70°

We also know that angles (Z + 10)° and X are opposite verticals, so we have:

Z + 10 = 70

Z = 60°

EXAMPLE 4

We have the angles (5x-11)° and (3x+23)°, which are opposite vertical angles. Determine the value of x and the size of the given angles.

Opposite vertical angles are equal, so we have:

5x-11 = 3x + 23

5x-3x = 23 + 11

2x = 34

x = 17

The value of the given angles is:

3(17)+23 = 74°


Vertical angles theorem – Practice problems

Solve the following problems using the vertical angles theorem. If you need help with this, look at the solved examples above.

What is the size of angles x and y?

ejercicio 5 de teorema de ángulos verticales

Choose an answer






Angles (3x-12)° and (2x+25)° are vertical angles. What is the value of x?

Choose an answer






If angles (4x-15)° and (3x+22)° are vertical angles, what is the size of one of the angles?

Choose an answer







See also

Interested in learning more about vertical angles and other types of angles? Take a look at these pages:

Learn mathematics with our additional resources in different topics

LEARN MORE