6.2: Identify Adjacent and Vertical Angles
Do you know about adjacent and vertical angles? Take a look at this illustration.
If you were asked to name all of the vertical angles and adjacent angles would you know what to identify? This Concept will teach you how to identify these special angle relationships.
Guidance
When lines intersect, they create special relationships between the angles that they form. Once we understand these relationships, we can use them to find the measure of angles formed by the intersecting lines.
Adjacent angles are angles that share the same vertex and one common side. If they combine to make a straight line, adjacent angles must add up to
Below, angles 1 and 2 are adjacent. Angles 3 and 4 are also adjacent.
Can you see that angles 1 and 2, whatever their measurements are, will add up to
We can find the sums in this way.
As you work through this lesson, you will find that some information leads you to other information. Here is the first example of that. Because adjacent angles form a straight line, they are also supplementary. The sum of the angles is
Notice that when there are two angles next to each other, there are also two angles diagonally across from each other. These are called vertical angles. Vertical angles are angles that are diagonally across from each other and have the same measure.
These relationships always exist whenever any two lines intersect. Look carefully at the figures below. Understanding the four angles formed by intersecting lines is a very important concept in geometry.
Identify the vertical angles and the adjacent angles in the diagram below.
First, think back to the definition of adjacent and vertical angles.
Adjacent angles form a straight line. They are supplementary angles. We can see from the diagram that angles 1 and 3 are adjacent. Angles 2 and 4 are also adjacent.
Vertical angles are diagonal from each other and have the same measure. In this case, angles 1 and 4 are vertical. Angles 2 and 3 are also vertical angles.
Example A
True or false. Adjacent angles are across from each other.
Solution: False. Adjacent angles are next to each other.
Example B
True or false. Pairs of vertical angles have the same measure.
Solution: True.
Example C
True or false. Adjacent angles can also be vertical angles.
Solution: False.
Now let's go back to the dilemma from the beginning of the Concept. Here is the illustration once again.
First identify the vertical angles.
Angles 1 and 3 are vertical angles. Angles 2 and 4 are also vertical angles.
Next, identify the adjacent angles. Remember that adjacent angles have a sum of 180^{\circ}.
Angles 1 and 2, angles 1 and 4, angles 4 and 3, and angles 2 and 3 are all adjacent angles.
Vocabulary
 Intersecting lines
 lines that cross at one point.
 Adjacent Angles

angles that are next to each other and whose sum is
180∘ .
 Vertical Angles

angles that are diagonally across from each other and whose sum is
90∘ .
 Angle
 the measure of the space formed by two intersecting lines.
Guided Practice
Here is one for you to try on your own.
Find
Solution
First, let's think about what we know about the angles identified.
This is our answer.
Video Review
Khan Academy Proving Vertical Angles are Equal
Practice
Directions: Identify whether each angle pair can be classified as adjacent angles or vertical angles or neither.
1.
2.
3.
4.
5.
6. If
Directions: Use this diagram to answer the following questions.
7. True or false.
8. What is the measure of
9. What is the measure of
10. What is the relationship between
11. True or false. Adjacent angles 1 and 2 form a straight line with a value of
Directions: Answer true or false for each question.
12. Supplementary angles are also vertical angles.
13. Vertical angles have the same measure.
14. Adjacent angles always have a sum of
15. Adjacent angles are also vertical angles.
16. Adjacent angles are formed when lines intersect.
Adjacent Angles
Two angles are adjacent if they share a side and vertex. The word 'adjacent' means 'beside' or 'nextto'.Angle
A geometric figure formed by two rays that connect at a single point or vertex.Diagram
A diagram is a drawing used to represent a mathematical problem.Intersecting lines
Intersecting lines are lines that cross or meet at some point.linear pair
Two angles form a linear pair if they are supplementary and adjacent.Vertical Angles
Vertical angles are a pair of opposite angles created by intersecting lines.Image Attributions
Here you'll identify adjacent and vertical angles formed by intersecting lines.