Triangle:

In geometry, triangle is one of the basic shapes. Triangle is the smallest polygon. It consists of three edges and three vertices. A triangle with vertices P, Q and R is denoted as ∆PQR. In a triangle, three sides and three angles are referred as the elements of the triangle. Angle sum property and exterior angle property are the two important attributes of a triangle. What is angle sum property of a triangle? How to prove angle sum property? It is discussed here on.

A triangle consists of interior and exterior angles. Interior angle is defined as the angle formed between two adjacent sides of a triangle. Exterior angle is defined as the angle formed between a side of triangle and an adjacent side extending outward.

For ∆ABC, Angle sum property of triangle declares that

Sum of all the interior angles of the triangle is 180°.

That is,

$m\angle{A}~+~m\angle{B}~+~m\angle{C}$ = $180^{\circ}$

Proof of Angle Sum Property Theorem

These are the following steps involved to prove angle sum property theorem:

Step 1: Draw a line AB through the vertex P and parallel to the side QR of a triangle PQR.

Step 2: Now, $\angle{APQ}~+~\angle{QPR}~+~\angle{RPB}$ = $180^{\circ}$[Linear Pair Axiom] —— (1)

Step 3:$\angle{APQ}$ = $\angle{PQR}$ [Alternate interior angles] —– (2)

Step 4: $\angle{RPB}$ = $\angle{PRQ}$  [Alternate interior angles] —– (3)

Step 5: Substituting $∠APQ$ and $∠RPB$ in equation 1 by $∠PQR$ and $∠PRQ$ respectively.