Vertical Angles

In these lessons, we will learn

  • how to identify vertical angles,
  • the vertical angle theorem,
  • how to solve problems involving vertical angles,
  • how to proof vertical angles are equal.

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Pairs Of Angles
Types Of Angles
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Vertical Angles

In geometry, pairs of angles can relate to each other in several ways. When two lines intersect, the opposite angles form vertical angles or vertically opposite angles. They are called vertical angles because they share the same vertex.

The Vertical Angle Theorem states that
Vertical angles are equal.

Notice also that x and y are supplementary angles i.e. their sum is 180°.

The following diagram shows the vertical angles formed from two intersecting lines. Scroll down the page for more examples and solutions.

vertical angles

The following diagram shows another example of vertical angles.

vertical angles

The following video explains more about vertical angles.

How to define and identify vertical angles?

A group of examples that identifies vertical angles.

Solving Problems using Vertical Angles

Very often math questions will require you to work out the values of angles given in diagrams by applying the relationships between the pairs of angles.

Given the diagram below, determine the values of the angles x, y and z.

vertical angles

Step 1: x is a supplement of 65°.
Therefore, x + 65° = 180° ⇒ x = 180° – 65° = 115°

Step 2: z and 115° are vertical angles.
Therefore, z = 115°

Step 3: y and 65° are vertical angles.
Therefore, y = 65°

Answer: x = 115°, y = 65° and z = 115°

Both AEC and DEB are straight lines.  Find q.


∠AEB = ∠DEC    ← vertical angles
 q + 45˚= 135˚
   q = 135˚ – 45˚ = 90˚

The following video shows how to find a missing vertical angle in a triangle.

Proof of the Vertical Angle Theorem

The following videos will prove that vertical angles are equal.

Try the free Mathway calculator and problem solver below to practice various math topics. Try the given examples, or type in your own problem and check your answer with the step-by-step explanations.
Mathway Calculator Widget

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