8.7: Triangle Sum Theorem
What if you knew that two of the angles in a triangle measured ? How could you find the measure of the third angle? After completing this Concept, you'll be able to apply the Triangle Sum Theorem to solve problems like this one.
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James Sousa: Animation of the Sum of the Interior Angles of a Triangle
Now watch this video.
James Sousa: Proving the Triangle Sum Theorem
Guidance
The Triangle Sum Theorem says that the three interior angles of any triangle add up to .
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Here is one proof of the Triangle Sum Theorem.
Given : with
Prove :
Statement | Reason |
---|---|
1. with | Given |
2. | Alternate Interior Angles Theorem |
3. | angles have = measures |
4. | Linear Pair Postulate |
5. | Angle Addition Postulate |
6. | Substitution PoE |
7. | Substitution PoE |
You can use the Triangle Sum Theorem to find missing angles in triangles.
Example A
What is ?
We know that the three angles in the triangle must add up to . To solve this problem, set up an equation and substitute in the information you know.
Example B
What is the measure of each angle in an equiangular triangle?
To solve, remember that is an equiangular triangle, so all three angles are equal. Write an equation.
If , then and .
Each angle in an equiangular triangle is .
Example C
Find the measure of the missing angle.
We know that and because it is a right angle. Set up an equation like in Example A.
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Guided Practice
1. Determine in this triangle:
2. Two interior angles of a triangle measure and . What is the third interior angle of the triangle?
3. Find the value of and the measure of each angle.
Answers:
1. .
Solve this equation and you find that .
2. .
Solve this equation and you find that the third angle is .
3. All the angles add up to .
Substitute in 12 for to find each angle.
Explore More
Determine in each triangle.
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8. Two interior angles of a triangle measure and . What is the third interior angle of the triangle?
9. Two interior angles of a triangle measure and . What is the third interior angle of the triangle?
10. Two interior angles of a triangle measure and . What is the third interior angle of the triangle?
Find the value of and the measure of each angle.
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Image Attributions
Description
Learning Objectives
Here you'll learn how to use the Triangle Sum Theorem, which states that the sum of the angles in any triangle is .