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.

### Watch This

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 .

.

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.

### 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.

### Practice

Determine in each triangle.

1.

2.

3.

4.

5.

6.

7.

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.

11.

12.

13.

14.

15.