 # An Introduction To Angle Sum Property Of A 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 to 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,

We have thus proved the angle sum property of a triangle. For the complete understanding of the topic please visit www.byjus.com or download the BYJU’S– The Learning App.

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