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. After the lesson, test yourself with a quiz.
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.
Here, we see two vapor trails that intersect. Therefore, they have created the pair of vertical angles labeled as 1 and 2.
Here is a pair of vertical angles formed in nature and that are more terrestrial.
If we draw a pair of intersecting lines, we have created two pairs of vertical angles. Here, angles AOC and BOD are a pair of vertical angles.
Angles AOB and COD are also a pair of 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. Here, each pair of vertical angles share vertex O.
Vertical Angles: More Examples
Let’s look at some more examples of vertical 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 62 and 53 and 84 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. Here, if we add in the angle measures, we’ll see that vertical angles are congruent.
Let’s do a simple proof for this. 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. Here is our 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).QED (our proof is complete)
Finding Angle Measures
Let’s now complete a problem! If angle 1 is 115 degrees, what are the measures of the other angles?The measure of angle 3 is 115 degrees because angles 1 and 3 are a pair of vertical angles. Angles 1 and 2 are a linear pair, so their sum is 180 degrees; therefore, the measure of angle 2 is 180 – 115 = 65 degrees. The measure of angle 4 is 65 degrees because angles 2 and 4 are a pair of vertical angles.
What Vertical Angles Are Not
Let’s finish this lesson by showing another non-example of vertical angles. Here, angles 1 and 3 are not a pair of vertical angles.
Even though they share a vertex and are not adjacent, they are not formed by the same pair of intersecting lines. Angle 1 is formed by lines r and t while angle 3 is formed by lines s and t.
Whenever two lines intersect, they form two pairs of vertical angles.
Vertical angles have a common vertex, but they are never adjacent angles. Finally, vertical angles are always congruent.
Vertical angles – a pair of non-adjacent angles formed when two lines intersectAdjacent angles – two angles which share a sideCongruent angles – angles which have the same angle measureLinear pair – a pair of adjacent anglesSupplementary – angles whose sum is 180 degrees
After this video, you should be able to:
- Define vertical angles
- Differentiate between vertical and adjacent angles
- Describe the properties of vertical angles
- Find the measure of an angle based on your knowledge of vertical angles