Chapter 4.1, Problem 14PSB

To determine

To Verify: $\angle P=\angle Q$

Explanation of Solution

Given information: T is midpoint of MN, $\angle PMT=\angle QNT={90}^{\circ },MR=SN,\angle 1=\angle 2.$

Formula Used: Congruency of triangles.

Proof:

  $\angle 1=\angle 2,RM=SN$ (Given)

  $\begin{array}{l}\therefore {90}^{\circ }-\angle 1={90}^{\circ }-\angle 2\\ \therefore \angle RMT=\angle SNT.\end{array}$

  $MT=TN$ (Given T is midpoint of MN).

  $\begin{array}{l}\Delta RMT\cong \Delta SNT(SAS)\\ \therefore \angle RTM=\angle STN(CPCT)\\ \therefore \Delta PMT\cong \Delta QNT(ASA)\\ \therefore \angle P=\angle Q(CPCT)\end{array}$