9.5 Triangle Sum Theorem
Learning Objectives
- I can use the Triangle Sum Theorem, which states that the sum of the angles in any triangle is .
Activity
Introduction
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.
Guided Learning
The Triangle Sum Theorem says that the three interior angles (or inside angles) of any triangle add up to .
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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 of this right triangle. Can you name the hypotenuse of this triangle? The hypotenuse is the longest side of a right triangle, the side opposite of the right angle. The other two sides of a right triangle are called legs.
We know that and because it is a right angle. Set up an equation like in Example A.
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.
Check your work with a partner.
Solutions
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.
Review
- You can use the Triangle Sum Theorem to find missing angle measurements in triangles.
Triangle Sum Theorem
The Triangle Sum Theorem states that the three interior angles of any triangle add up to .
Interior angles
The inside angles connecting the sides of the polygon are called interior angles.
Hypotenuse
The hypotenuse is the longest side of a right triangle, the side opposite of the right angle.
Leg
The two sides of a right triangle that form the right angle are called the legs of that triangle.
Additional Resources
Sum of Interior Angles of a Triangle Video
Proving Triangle Sum Theorem Video