4.2: Exterior Angles Theorems
What if you knew that two of the exterior angles of a triangle measured ? How could you find the measure of the third exterior angle? After completing this Concept, you'll be able to apply the Exterior Angle Sum Theorem to solve problems like this one.
Watch This
CK-12 Foundation: Chapter4ExteriorAnglesTheoremsA
James Sousa: Introduction to the Exterior Angles of a Triangle
James Sousa: Proof that the Sum of the Exterior Angles of a Triangle is 360 Degrees
James Sousa: Proof of the Exterior Angles Theorem
Guidance
An exterior angle is the angle formed by one side of a polygon and the extension of the adjacent side. In all polygons, there are two sets of exterior angles, one going around the polygon clockwise and the other goes around the polygon counterclockwise. By the definition, the interior angle and its adjacent exterior angle form a linear pair.
The Exterior Angle Sum Theorem states that each set of exterior angles of a polygon add up to .
Remote interior angles are the two angles in a triangle that are not adjacent to the indicated exterior angle. and are the remote interior angles for exterior angle .
The Exterior Angle Theorem states that the sum of the remote interior angles is equal to the non-adjacent exterior angle. From the picture above, this means that . Here is the proof of the Exterior Angle Theorem. From the proof, you can see that this theorem is a combination of the Triangle Sum Theorem and the Linear Pair Postulate.
Given : with exterior angle
Prove :
Statement | Reason |
---|---|
1. with exterior angle | Given |
2. | Triangle Sum Theorem |
3. | Linear Pair Postulate |
4. | Transitive PoE |
5. | Subtraction PoE |
Example A
Find the measure of .
is an exterior angle of . Therefore, it is supplementary to because they are a linear pair.
If we draw both sets of exterior angles on the same triangle, we have the following figure:
Notice, at each vertex, the exterior angles are also vertical angles, therefore they are congruent.
Example B
Find the measure of the numbered interior and exterior angles in the triangle.
by the Linear Pair Postulate, so .
by the Linear Pair Postulate, so .
by the Triangle Sum Theorem, so and .
by the Linear Pair Postulate, so .
Example C
What is the value of in the triangle below?
First, we need to find the missing exterior angle, we will call it . Set up an equation using the Exterior Angle Sum Theorem.
and are supplementary and add up to .
Watch this video for help with the Examples above.
CK-12 Foundation: Chapter4TriangleSumTheoremA
Concept Problem Revisited
The third exterior angle of the triangle below is .
By the Exterior Angle Sum Theorem:
Vocabulary
Interior angles are the angles on the inside of a polygon while exterior angles are the angles on the outside of a polygon. Remote interior angles are the two angles in a triangle that are not adjacent to the indicated exterior angle. Two angles that make a straight line form a linear pair and thus add up to . The Triangle Sum Theorem states that the three interior angles of any triangle will always add up to . The Exterior Angle Sum Theorem states that each set of exterior angles of a polygon add up to .
Guided Practice
1. Find .
2. Find .
3. Find the value of and the measure of each angle.
Answers:
1. Set up an equation using the Exterior Angle Theorem. . Therefore, .
2. Using the Exterior Angle Theorem, . Subtracting from both sides, .
3. Set up an equation using the Exterior Angle Theorem.
Substituting back in for , the two interior angles are and . The exterior angle is . Double-checking our work, notice that . If we had done the problem incorrectly, this check would not have worked.
Interactive Practice
Practice
Determine .
Use the following picture for the next three problems:
- What is ?
- What is ?
- What is ?
Solve for .
- Suppose the measures of the three angles of a triangle are x, y, and z. Explain why .
- Suppose the measures of the three angles of a triangle are x, y, and z. Explain why the expression represents the sum of the exterior angles of the triangle.
- Use your answers to the previous two problems to help justify why the sum of the exterior angles of a triangle is 360 degrees. Hint: Use algebra to show that must equal 360 if .