Question

In triangle ABC, the bisector of an interior angle at vertex B and the bisector of the exterior angle at vertex A intersect each other at point P. then, 2 $\angle APB=\angle C$.
State true or false.

• A. True
• B. False

Answer 1

The correct option is A True

$\angle ABP=\frac{1}{2}\angle B$ (BP bisects $\angle B$)
$\angle BAP=\angle A+\frac{1}{2}(Ext.\angle A)$ (AP bisects $\angle A$)
$\angle BAP=\angle A+\frac{1}{2}(180-\angle A)$
$\angle BAP=90+\frac{1}{2}\angle A$
In $△ABP$,
$\angle BAP+\angle PBA+\angle APB=180$ (Sum of angles of triangle)
$90+\frac{1}{2}\angle A+\frac{1}{2}\angle B+\angle APB=180$
$\angle APB=90-\frac{1}{2}(\angle A+\angle B)$
$2\angle APB=180-\angle A-\angle B$
$2\angle APB=\angle C$ (Angles sum property on triangle ABC)