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Vertical Angles in Geometry: Definition, Examples & Quiz

Instructor: David Liano

After completing this lesson, you will be able to identify and draw vertical angles. You will also be able to state the properties of vertical angles.

We also recommend watching Types of Angles: Vertical, Corresponding, Alternate Interior & Others and Angles and Triangles: Practice Problems

Definition: Vertical Angles

Vertical angles are a pair of non-adjacent angles formed when two lines intersect. We see intersecting lines all the time in our real world. In Figure 1, we see two vapor trails that intersect. Therefore, they have created the pair of vertical angles labeled as 1 and 2.

Figure 1
vapor trails

Figure 2 shows a pair of vertical angles formed in nature and that are more terrestrial.

Figure 2
leaves

If we draw a pair of intersecting lines, we have created two pairs of vertical angles. In Figure 3, angles AOC and BOD are a pair of vertical angles. Angles AOB and COD are also a pair of vertical angles.

Figure 3
vertical angles

Notice that vertical angles are never adjacent angles. In other words, they never share a side. For example, angles AOC and AOB are not a pair vertical angles, but they are adjacent angles. However, vertical angles always have a common vertex. In Figure 3, each pair of vertical angles share vertex O.

Vertical Angles: More Examples

Let's look at some more examples of vertical angles. We will use the diagram shown in Figure 4.

Figure 4
types of angles

Line c intersects two lines, a and b. Vertical angles are formed at each intersection. The vertical pairs of angles are as follows:

1 and 6

2 and 5

3 and 8

4 and 7

Vertical Angles: Congruency Property

A primary property of vertical angles is that they are congruent. In other words, they have the same angle measure. Let's use the same drawing from Figure 3. If we add in the angle measures, we will see that vertical angles are congruent.

Vertical Angles Are Congruent
vertical angles

Proof

Let's do a simple proof for the above property. Before we begin, we should acknowledge some definitions and theorems in geometry. First of all, a linear pair of angles is a pair of adjacent angles. Their non-common sides are always opposite rays. In addition, angles that form a linear pair are also supplementary, so their sum is always 180 degrees.

We will use the drawing shown in Figure 5 for our proof:

Figure 5
proof

1. Lines m and n intersect forming angles 1, 2, 3, and 4 (given).

2. Angles 1 and 2 are a linear pair, so they are supplementary (definition of linear pair).

3. Angle 1 + angle 2 = 180 degrees (definition of supplementary angles).

4. Angles 2 and 3 are a linear pair, so they are supplementary (definition of linear pair).

5. Angle 2 + angle 3 = 180 degrees (definition of supplementary angles).

6. Angle 1 + angle 2 = angle 2 + angle 3 (substitution; see statements 3 and 5).

7. Angle 1 = angle 3 (subtract angle 2 from the equation in statement 6).

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