Chapter 8, Problem 17RP

To determine

To prove:The given ratio is congruent to each other i.e. $\frac{\text{MO}}{\text{OP}}\cong \frac{\text{MT}}{\text{SR}}$

Explanation of Solution

Given information:

  $\begin{array}{l}\overline{\text{OS}}\parallel \overline{\text{PR}}\\ \angle 1\cong \angle 2\end{array}$

Formula used:

Congruency of triangles

Vertical angles are always equal

Proof :

In Triangles MOT and RST, we have

  $\begin{array}{l}\angle 1=\angle \text{MTO (vertical angles)}\\ \angle \text{2=}\angle \text{MOT (alternate angles)}\\ \text{OP=SR (since OS}\parallel \text{PR)}\\ \text{OP=OT}\\ \therefore \text{OT=SR}\\ \Rightarrow \Delta \text{MOT}\cong \Delta \text{SRT}\end{array}$

→ASA criteria of congruency is satisfied)

Hence, ratio of corresponding sides will be equal

  $\frac{\text{MO}}{\text{OP}}\text{=}\frac{\text{MT}}{\text{SR}}$