Chapter 5.2, Problem 21PSA

To determine

To Prove: $\overline{MO}\parallel \overline{RP}$ .

Explanation of Solution

Given:

  

  $\begin{array}{l}\angle MOP={90}^{\circ }\\ \overline{RP}\perp \overline{OP}\end{array}$

Concept Used:

When parallel lines is intersected by a interior angles on the same side of transversal are supplementary.

Proof:

Here, we have

  $\begin{array}{l}\angle MOP={90}^{\circ }\\ \overline{RP}\perp \overline{OP}\\ \Rightarrow \angle RPO={90}^{\circ }\\ \therefore \angle MOP+\angle RPO={180}^{\circ }\\ \Rightarrow \angle MOP\text{and }\angle RPO\text{are supplementary}\\ \text{Since, we know that when parallel lines is intersected by transversal interior angles}\\ \text{on the same side of transversal are supplementary}\text{.}\\ \Rightarrow \overline{MO}\parallel \overline{RP}\text{with transversal }\overline{OP}\end{array}$