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

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

Related Topics: More Geometry Lessons

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:

This means that:

(i)qandsare vertical angles

(ii)q=sSimilarly,

pis opposite tor

This means that:

(i)pandrare vertical angles

(ii)p=rNotice also that

pandqare supplementary angles i.e. their sum is 180°. Similarly,sandrare supplementary angles.

The following diagram shows another example of vertical angles.

A group of examples that identifies 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**.*

**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.