Exterior Angle Property Of A Triangle

Theorem: The exterior angle at a vertex of a triangle equals the sum of the interior angles at the other two vertices of the triangle.

Prerequisites:
Angle sum property of a triangle (proof)
Angle on a straight line (proof)

Proof:

In the given triangle, $\angle CBD$ is an exterior angle at the vertex B. Since AD is a straight line,

$\therefore\quad\;\;\angle CBD + \angle ABC = 180^o$
$\Rightarrow\quad\;\;\angle CBD = 180^o - \beta$

Also, by the angle sum property of a triangle,

$\qquad\quad\angle A + \angle ABC + \angle C = 180^o$
$\begin{align}\Rightarrow\quad\;\;\angle A + \angle C &= 180^o - \beta\\ &= \angle CBD\end{align}$

Hence the result.


Recommended:
Angle sum property of polygon
Isosceles Triangle Theorem
Interior angles property