# 4.17: Triangle Angle Sum Theorem

- Page ID
- 4814

The interior angles of a triangle add to 180 degrees Use equations to find missing angle measures given the sum of 180 degrees.

## Triangle Sum Theorem

The **Triangle Sum Theorem** says that the three interior angles of any triangle add up to \(180^{\circ}\).

\(m\angle 1+m\angle 2+m\angle 3=180^{\circ}\).

Here is one proof of the Triangle Sum Theorem.

__Given__: \(\Delta ABC\) with \(\overleftrightarrow{AD} \parallel \overline{BC}\)

__Prove__: \(m\angle 1+m\angle 2+m\angle 3=180^{\circ}\)

Statement |
Reason |
---|---|

1. \(\Delta ABC with \overleftrightarrow{AD} \parallel \overline{BC}\) | Given |

2. \\(angle 1\cong \angle 4,\: \angle 2\cong \angle 5\) | Alternate Interior Angles Theorem |

3. \(m\angle 1=m\angle 4,\: m\angle 2=m\angle 5\) | \cong angles have = measures |

4. \(m\angle 4+m\angle CAD=180^{\circ}\) | Linear Pair Postulate |

5. \(m\angle 3+m\angle 5=m\angle CAD\) | Angle Addition Postulate |

6. \(m\angle 4+m\angle 3+m\angle 5=180^{\circ}\) | Substitution PoE |

7. \(m\angle 1+m\angle 3+m\angle 2=180^{\circ}\) | Substitution PoE |

You can use the Triangle Sum Theorem to find missing angles in triangles.

What if you knew that two of the angles in a triangle measured \(55^{\circ}\)? How could you find the measure of the third angle?

Example \(\PageIndex{1}\)

Two interior angles of a triangle measure \(50^{\circ}\) and \(70^{\circ}\). What is the third interior angle of the triangle?

**Solution**

\(50^{\circ}+70^{\circ}+x=180^{\circ}\).

Solve this equation and you find that the third angle is \(60^{\circ}\).

Example \(\PageIndex{2}\)

Find the value of \(x\) and the measure of each angle.

**Solution**

All the angles add up to \(180^{\circ}\).

\(\begin{align*} (8x−1)^{\circ}+(3x+9)^{\circ}+(3x+4)^{\circ}&=180^{\circ} \\ (14x+12)^{\circ}&=180^{\circ} \\ 14x&=168 \\ x&=12\end{align*} \)

Substitute in 12 for \(x\) to find each angle.

\([3(12)+9]^{\circ}=45^{\circ} \qquad [3(12)+4]^{\circ}=40^{\circ} \qquad [8(12)−1]^{\circ}=95^{\circ}\)

Example \(\PageIndex{3}\)

What is m\angle T?

**Solution**

We know that the three angles in the triangle must add up to \(180^{\circ}\). To solve this problem, set up an equation and substitute in the information you know.

\(\begin{align*} m\angle M+m\angle A+m\angle T&=180^{\circ} \\ 82^{\circ}+27^{\circ}+m\angle T&=180^{\circ} \\ 109^{\circ}+m\angle T&=180^{\circ} \\ m\angle T &=71^{\circ}\end{align*}\)

Example \(\PageIndex{4}\)

What is the measure of each angle in an equiangular triangle?

**Solution**

To solve, remember that \(\Delta ABC\) is an equiangular triangle, so all three angles are equal. Write an equation.

\(\begin{align*} m\angle A+m\angle B+m\angle C &=180^{\circ} \\ m\angle A+m\angle A+m\angle A&=180^{\circ} \qquad &Substitute,\: all\: angles\: are \: equal. \\ 3m\angle A&=180^{\circ} \qquad &Combine\:like \:terms. \\ m\angle A&=60^{\circ}\end{align*}\)

If \(m\angle A=60^{\circ}\), then \(m\angle B=60^{\circ}\) and \(m\angle C=60^{\circ}\).

Each angle in an equiangular triangle is \(60^{\circ}\).

Example \(\PageIndex{5}\)

Find the measure of the missing angle.

**Solution**

We know that \(m\angle O=41^{\circ}\) and \(m\angle G=90^{\circ}\) because it is a right angle. Set up an equation like in Example 3.

\(\begin{align*} m\angle D+m\angle O+m\angle G&=180^{\circ} \\ m\angle D+41^{\circ}+90^{\circ}&=180^{\circ} \\ m\angle D+41^{\circ}&=90^{\circ}\\ m\angle D=49^{\circ}\end{align*}\)

## Review

Determine \(m\angle 1\) in each triangle.

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2.

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6.

7.

8. Two interior angles of a triangle measure \(32^{\circ}\) and \(64^{\circ}\). What is the third interior angle of the triangle?

9. Two interior angles of a triangle measure \(111^{\circ}\) and \(12^{\circ}\). What is the third interior angle of the triangle?

10. Two interior angles of a triangle measure \(2^{\circ}\) and \(157^{\circ}\). What is the third interior angle of the triangle?

Find the value of \(x\) and the measure of each angle.

11.

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15.

## Review (Answers)

To see the Review answers, open this PDF file and look for section 4.1.

## Resources

## Vocabulary

Term | Definition |
---|---|

Triangle Sum Theorem |
The Triangle Sum Theorem states that the three interior angles of any triangle add up to 180 degrees. |

## Additional Resources

Interactive Element

Video: Triangle Sum Theorem Principles - Basic

Activities: Triangle Sum Theorem Discussion Questions

Study Aids: Triangle Relationships Study Guide

Practice: Triangle Angle Sum Theorem

Real World: Triangle Sum Theorem