Chapter 3.7, Problem 9PSA

To determine

To Prove: Congruency of the $\overline{MP}$ with respect to $\overline{TR}$

Explanation of Solution

Given information: $\overline{OP}\cong \overline{RS}$ , $\overline{KO}\cong \overline{KS}$ , M is the mid-point of $\overline{OK}$ and T is the mid-point of $\overline{KS}$ is provided in the question which will help to find the solution.

Proof: Since, $\overline{KO}\cong \overline{KS}$

So that, $\angle KOS=\angle KSO$ (opposite angles of equal sides are equal)

And, $\overline{OP}\cong \overline{RS}$

So, $\angle OMP=\angle RTS$

So that, $\overline{OP}\cong \overline{RS}$ (given)

  $\angle KOS=\angle KSO$ (proved above)

  $\angle OMP=\angle RTS$ (proved above)

  $\therefore$

  $\Delta OMP\cong \Delta RTS$

Hence, $\overline{MP}\cong \overline{TR}$ (by Corresponding Part of Congruent Triangle)