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# 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,

$$\begin{array}{l}m\angle{A}~+~m\angle{B}~+~m\angle{C}\end{array}$$
=
$$\begin{array}{l}180^{\circ}\end{array}$$

## 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,

$$\begin{array}{l}\angle{APQ}~+~\angle{QPR}~+~\angle{RPB}\end{array}$$
=
$$\begin{array}{l}180^{\circ}\end{array}$$
[Linear Pair Axiom] —— (1) Step 3:

$$\begin{array}{l}\angle{APQ}\end{array}$$
=
$$\begin{array}{l}\angle{PQR}\end{array}$$
[Alternate interior angles] —– (2) Step 4:

$$\begin{array}{l}\angle{RPB}\end{array}$$
=
$$\begin{array}{l}\angle{PRQ}\end{array}$$
[Alternate interior angles] —– (3) Step 5: Substituting

$$\begin{array}{l}∠APQ\end{array}$$
and
$$\begin{array}{l}∠RPB\end{array}$$
in equation 1 by
$$\begin{array}{l}∠PQR\end{array}$$
and
$$\begin{array}{l}∠PRQ\end{array}$$
respectively.

We have thus proved the angle sum property of a triangle.

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