Vertical Angles

In geometry, pairs of angles can relate to each other in several ways.
In this lesson, 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

Related Topics: More Geometry Lessons

Vertical Angles

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.

Example:
vertical angles

This means that:
 (i)  q and s are vertical angles
 (ii) q = s

Similarly, p is opposite to r
This means that:
 (i)  p and r are vertical angles
 (ii) p = r

Notice also that p and q are supplementary angles i.e. their sum is 180°. Similarly, s and r are supplementary 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.

Example:

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

vertical angles

Solution:

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°

 

Example:

Both AEC and DEB are straight lines.  Find q.

 

Solution:

   ← vertical angles
 q + 45˚= 135˚
    q = 135˚ – 45˚ = 90˚

The following video shows how to use the vertical angle theorem to solve problems.
Identify vertical angles and find the missing angle measures from a diagram.




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.









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