Question

a. given : $overline{pr}$ bisects $angle apb$, $angle apb$ is a right - angle. prove : $mangle rpb = 45^{circ}$

Explanation:

Step1: Recall angle - bisector definition

An angle - bisector divides an angle into two equal parts. Given that $\overrightarrow{PR}$ bisects $\angle APB$, we have $\angle APR=\angle RPB$.

Step2: Use the measure of $\angle APB$

Since $\angle APB$ is a right - angle, $m\angle APB = 90^{\circ}$. And because $\angle APR+\angle RPB=\angle APB$ and $\angle APR = \angle RPB$, we can substitute to get $2m\angle RPB=m\angle APB$.

Step3: Solve for $m\angle RPB$

Substitute $m\angle APB = 90^{\circ}$ into the equation $2m\angle RPB=m\angle APB$. Then $2m\angle RPB = 90^{\circ}$, and $m\angle RPB=\frac{90^{\circ}}{2}=45^{\circ}$.

Answer:

$m\angle RPB = 45^{\circ}$