Triangle Sum Theorem

What is the Triangle Sum Theorem

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°

Prove Triangle Sum Theorem

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°.

Solved Examples

Let us solve some problems to understand the theorem better.

Corollary to the Triangle Sum Theorem

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°