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Last modified on August 3rd, 2023

The triangle sum theorem states that the sum of the three interior angles in a triangle adds up to 180°. It is also called the angle sum theorem.

Given below is a triangle ABC, having three interior angles ∠a, ∠b, and ∠c. According to the triangle sum theorem, ∠a + ∠b + ∠c = 180°

**To prove:**

**∠CBA + ****∠BAC + ****∠ACB** **= 180****°**

**Proof**:

A line PQ is drawn parallel to BC passing through the point A

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

1. | ∠PAB + ∠BAC + ∠QAC = 180°……(1) | PQ is a straight line |

2. | ∠QAC = ∠ACB ……(2) | Pair of alternate interior angles. PQ ||BC, and AB, AC are transversals |

3. | ∠PAB = ∠CBA …….(3) | Pair of alternate interior angles. PQ ||BC, and AB, AC are transversals |

4. | ∠CBA + ∠BAC + ∠ACB = 180° | Substituting (2) and (3) in (1) |

Hence proved that, the sum of the three interior angles in a triangle adds up to 180°.

Let us solve some problems to understand the theorem better.

**Find the value of the unknown angle of the given triangle.**

Solution:

As we know, according to the triangle sum theorem,

x + 38° + 32° = 180°

=> x = 180° – (38° + 32°)

=> x = 110°

**Two interior angles of a triangle measure 30**°** and 80**°**. What is the third interior angle of the triangle?**

Solution:

Let the third interior angle be x

As we know, according to the triangle sum theorem,

x + 30° + 80° = 180°

x = 180° – (30° + 80°)

x = 70°

**Solve the value of x and the measure of each angle.**

Solution:

As we know, according to the triangle sum theorem,

(8x – 1)° + (4x + 8)° + (3x + 8)° = 180°

15x + 15 = 180°

15x = 165

x = 11°

Two corollaries to the triangle sum theorem are:

**Corollary 1**: The acute angles of a right triangle are complementary (add up to 90°)

Hypothesis: From the triangle sum theorem, the sum of all three angles equals 180°

Again, from the definition of a right triangle, we have one of its angles to be a right angle, making the remaining angles to be acute whose sum equals (180° – 90°) is 90°

Conclusion: The acute angles of a right triangle are complementary

**Corollary 2**: Each angle in an equilateral triangle measures 60°

Hypothesis: From the triangle sum theorem, the sum of all three angles equals 180°

Again, from the definition of an equilateral triangle, all angles are of equal measure. Adding up all the angles, we get,

⇒ x + x + x = 180°

⇒ 3x = 180°

⇒ x = 60°

Conclusion: Each angle in an equilateral triangle measures 60°

Last modified on August 3rd, 2023