Chapter 2.8, Problem 5PSA

To determine

The congruency of the given angles.

Explanation of Solution

Given information:

  $\angle 1\cong \angle 4$

Formula used :

Congruency means same shape and same size

Vertically opposite angles (VOAs) are always equal

Proof :

Since,

  $\angle 1\cong \angle 4$ ……..(Given)

  $\begin{array}{l}\therefore \angle \text{1=}\angle \text{4}\mathrm{........}\left(i\right)\\ \left(\because \text{same shape and same size by definition of congruency}\right)\\ \angle 1=\angle \mathrm{2.....}\left(\because \text{ both are vertically opposite angles}\right)\mathrm{......}\left(ii\right)\\ \angle 4=\angle \mathrm{3.....}\left(\because \text{ both are vertically opposite angles}\right)\mathrm{.......}\left(iii\right)\\ \text{From }\left(ii\right)\&\left(iii\right),\text{we have}\\ \angle \text{2=}\angle \text{3}\mathrm{........}\left(\because \text{ from }\left(i\right),\angle \text{1=}\angle \text{4}\right)\\ \text{Thus,}\\ \angle \text{2}\cong \angle \text{3}\mathrm{......}\left(\text{by definition of congruency}\right)\\ \text{Hence Prooved}\end{array}$