Vertical Angles Theorem – Definition, Proof and Examples

Vertical angles are formed at the intersection of two lines. The vertical angles theorem tells us that the vertical angles formed at an intersection are equal. Also, a vertical angle and its adjacent angle are supplementary angles since they add up to 180 degrees.

Here, we will look at detailed definitions of vertical angles and the vertical angles theorem. Then, we will learn how to derive this theorem and apply it to solve some practice problems.

GEOMETRY
diagram of opposite vertical angles

Relevant for

Learning about the vertical angles theorem.

See theorem

GEOMETRY
diagram of opposite vertical angles

Relevant for

Learning about the vertical angles theorem.

See theorem

What are the vertical angles?

Vertical angles are formed at the intersection of two lines. These angles are also commonly known as vertex angles.

Recall that intersecting lines are lines that cross each other at a specific point. The following is a diagram of intersecting lines:

intersecting lines

We can see that lines AB and CD meet at point P. Therefore, these lines are intersecting.

On the other hand, parallel lines are lines that never cross each other, as we have in the following diagram:

parallel lines

Parallel lines do not form vertical angles.

Two pairs of vertical angles are formed at the intersection of two lines, as in the following diagram:

diagram of opposite vertical angles

In this diagram:

  • ∠a and ∠b are vertical angles.
  • ∠c and ∠d are vertical angles.

What is the vertical angles theorem?

The vertical angles theorem tells us that, “the vertical angles formed by the intersection of two lines are equal.”

This means that the vertical angles are always equal to each other.

diagram of opposite vertical angles

Therefore, in this diagram, we have:

  • ∠a = ∠b.
  • ∠c = ∠d.

Proof of the vertical angles theorem

In the following diagram, we can see lines AB and CD, which intersect at point O. The intersection of lines forms two pairs of vertical angles:

  • ∠1 and ∠2 (blue angles)
  • ∠3 and ∠4 (pink angles)
vertical angles theorem

In the diagram, we see that ray OA is located on line CD. Therefore, we can use the axiom of linear pairs, which tells us that if a ray is located on a line, the adjacent angles form a linear pair of angles. That is, we know that the angles that form a straight line must add up to 180°. So, we have:

∠1 + ∠3 = 180° (linear pair)

Similarly, we also have:

∠3 + ∠2 = 180° (linear pair)

Since both equations are equal to 180°, we can combine them to obtain:

∠1 + ∠3 = ∠3 + ∠2

∠1 = ∠2

This means that the vertical angles ∠1 and ∠2 are equal. Therefore, we have proved the vertical angles theorem.


Examples of the vertical angles theorem

The following examples are solved by applying the vertical angles theorem.

EXAMPLE 1

Determine the sizes of angles a, b, and c in the following diagram:

opposite angles by the vertex example 1

Solution: We see that the angles ∠50 ° and ∠b are a pair of vertical angles, so we have:

∠b = 50°

Now, we can find the value of ∠a by considering that it is a supplementary angle to the angle ∠50 °, so we have:

50° + ∠a = 180°

∠a = 130°

The angles ∠a and ∠c are another pair of vertical angles, so we have:

∠c = 130°

EXAMPLE 2

Find the value of Y in the following diagram:

opposite angles by the vertex example 2

Solution: We can see that the angles ∠100° and ∠X form a straight line, so they are supplementary. Therefore, we have:

100° + ∠X = 180°

∠X = 80°

Also, we know that the angles (Y + 30)° and X are vertical opposite, so we have:

Y + 30 = 80

Y = 50°

EXAMPLE 3

We have that the angles (4x-15)° and (3x+22)° are opposite vertical angles. Determine the value of x and the size of the given angles.

Solution: The opposite vertical angles are equal, so we have:

4x-15 = 3x + 22

4x-3x = 22 + 15

x = 37

The value of the given angles is:

4(37)-15 = 133°


See also

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

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