# Geometry: Second Theorem

Geometry: Second Theorem

Published on ginnyent on 7th May, 2019

The second theorem says, “The exterior angle of a triangle is equal to the sum of the opposite interior angles”

How do we prove the above statement?

Given any triangle which could be identified using any letters of alphabet, let’s use ABC for convenience. See the figure below. Photo: Channon et al., 2001

To prove that the angle at C (outside the triangle) is equal to the sum of the angles at A and B (inside the triangle).

We extended BC to X by construction. To prove that angle ACX = angle at A + angle at B

From the figure above, ACX + ACB = 1800 (BCX is a straight line; sum of angles on a straight line is 1800)

Angle at A + angle at B + ACB (angle at C inside the triangle) = 1800

We have already proved that the sum of the angles in a triangle is 1800. See our first post, Formal Theorems of Geometry

It then follows that angle ACB (angle at C inside the triangle) = 180 – (angle A + angle  B)

Next: The Third Theorem