Question

You know the measure of the exterior angle which forms a linear pair with the vertex angle. Describe two ways you can find measures of the interior angles of the triangle.

Answer 1

Answer:

Step-by-step explanation:

A linear pair are two given angles whose sum is equal to $180^{o}$.

Let the exterior angle be represented by $x^{o}$, and the three interior angles of the triangle be represented by $a^{o}$, $b^{o}$ and $c^{o}$.

Assume that the vertex angle $c^{o}$ is the linear pair to the exterior angle $x^{o}$.

i.e            $x^{o}$ + $c^{o}$ = $180^{o}$  (sum of angles on a straight line)

Thus,

i. $a^{o}$ +  $b^{o}$ = $x^{o}$  (sum of two opposite interior angles is equal to an exterior angle)

⇒ $a^{o}$  = $x^{o}$ - $b^{o}$

Also,

$b^{o}$  = $x^{o}$ - $a^{o}$

But,

$a^{o}$ + $b^{o}$ + $c^{o}$ = $180^{o}$   (sum of angles in a triangle)

⇒ $c^{o}$ = $180^{o}$ - ($a^{o}$ +  $b^{o}$)

and

$a^{o}$ + $b^{o}$ = $180^{o}$ - $c^{o}$

ii. $a^{o}$ + $b^{o}$ + $c^{o}$ = $180^{o}$   (sum of angles in a triangle)

$a^{o}$ = $180^{o}$ - ($b^{o}$ + $c^{o}$)

Also,

$a^{o}$ +  $b^{o}$ = $x^{o}$

⇒ $b^{o}$ = $x^{o}$ - $a^{o}$

$c^{o}$ = $180^{o}$ - $x^{o}$