Chapter 4.4, Problem 11PSB

To determine

To Verify: $PS\perp QR.$

Explanation of Solution

Given information: $PQ=PR,QS\text{and}RS\text{are}\text{angle}\text{bisectors}\text{.}$ .

Formula used: congruency of triangles, linear pair axiom.

Proof:

  $\begin{array}{l}PQ=PR,QS\text{and}RS\text{are}\text{angle}\text{bisectors}\text{.}(\text{given})\\ \therefore \angle PQM=\angle PRM(\text{Angles opposite to equal sides are equal})\\ \therefore \frac{1}{2}\angle PQM=\frac{1}{2}\angle PRM.\\ \therefore \angle PQS=\angle SQM=\angle SRM=\angle PRM.\\ PS=PS(\text{Reflective}\text{property)}\\ QS=RS\text{(Sides opposite to equal angles are equal)}\\ \therefore \Delta PQS\cong \Delta PRS(\text{SSS})\\ \therefore \angle QPM=\angle RPM(\text{CPCT})\\ PM=PM(\text{Reflective}\text{property)}\\ \therefore \Delta PQM\cong \Delta PRM(\text{SAS})\\ \therefore \angle PMQ=\angle PMR(CPCT)\\ \angle PMQ+\angle PMR={180}^{\circ }(\text{Linear pair axiom})\\ \therefore \angle PMQ={90}^{\circ }=\angle PMR.\end{array}$