Chapter 2.8, Problem 10PSB

To determine

To prove that the two given angles are congruent.

Answer to Problem 10PSB

  $\angle \text{V is congruent to }\angle \text{Y }$

Explanation of Solution

Given information :

  $\begin{array}{l}\angle \text{V is congruent to }\angle \text{YRX and}\\ \angle \text{Y is congruent to }\angle \text{TRV}\end{array}$

Formula used :

Same shape and same size mean congruent

Vertically opposite angles are equal Proof :

  $\begin{array}{l}\text{Since, given}\\ \angle \text{V }\cong \angle \text{YRX and }\angle \text{Y }\cong \angle \text{TRV}\\ \therefore \angle \text{V = }\angle \text{YRX and }\angle \text{Y= }\angle \text{TRV}\mathrm{.......}\left(i\right)\\ \left(\text{same shape and size by definintion of congruency}\right)\end{array}$

From figure, it is clear that

  $\begin{array}{l}\angle \text{YRX and }\angle \text{TRV are vertical angles}\\ \therefore \angle \text{YRX=}\angle \text{TRV}\mathrm{...........}\left(ii\right)\left(\text{vertical angles are always equal}\right)\end{array}$

From (i) and (ii), we get

  $\begin{array}{l}\angle \text{V = }\angle \text{Y}\\ \text{Thus, }\angle \text{V }\cong \angle \text{Y}\mathrm{......}\left(\text{by definition of congruency}\right)\end{array}$

Hence, Proved