Chapter 3.5, Problem 7PSB

To determine

To prove:That the given line segments are congruent.

Explanation of Solution

Given information In a given figure

NOPRS is an equilateral

  $\angle \text{OPR }\cong \angle \text{PRS}$

  $\overline{\text{PT}}\cong \overline{\text{TR}}$

Formula used Definition of congruency means same size and shame shape

SAS criteria of congruency

Congruent parts of congruent triangle are congruent

Proof

  $\begin{array}{l}\text{In }\Delta \text{OPT and }\Delta \text{SRT, we have}\\ \overline{\text{PT}}\cong \overline{\text{TR}}\mathrm{.......}\text{given}\\ \overline{\text{PT}}=\overline{\text{TR}}\mathrm{........}\text{by definition of congruency}\mathrm{......}\left(i\right)\\ \overline{\text{OP}}=\overline{\text{SR}}\mathrm{......}\text{given sides are congruent}\mathrm{......}\left(ii\right)\\ \angle \text{OTP=}\angle \text{STR}\mathrm{.....}\text{ vertical angles}\mathrm{........}\left(iii\right)\\ From\left(i\right),\left(ii\right)\text{and}\left(iii\right),\text{we have}\\ \Delta \text{OPT }\cong \Delta \text{SRT}\mathrm{......}\text{SAS criteria of congruency satisfied}\\ \text{⇒}\overline{\text{OT}}=\overline{\text{ST}}\mathrm{.......}\text{Congruent parts of congruent triangle are congruent}\\ \text{Hence proved}\end{array}$