Chapter 5, Problem 30CUR

To determine

To prove $\angle WXY\cong \angle ZYX$ .

Explanation of Solution

Given:

  

Proof:

In $\Delta WPX$ and $\Delta ZPY$

  $\angle WPX=\angle ZPY$ (vertically opposite angles are equal)

  $WP=ZP$ (given)

  $PX=PY$ (given)

  $\therefore \Delta WPX\cong \Delta ZPY$ by Side-Angle-Side congruency criteria.

  $\Rightarrow \angle WXP\cong \angle ZYP$ (Corresponding parts of congruent triangles are congruent)….(i).

Now, in $\Delta XPY$

  $XP=PY$

  $\Rightarrow \angle PXY=\angle PYX$ (angles opposite to equal sides are equal)….(ii).

Adding equation (i) and equation (ii),

  $\angle WXP+\angle PXY=\angle ZYP+\angle PYX$

  $\Rightarrow \angle WXY=\angle ZYX$

Hence Proved.

Conclusion:

Therefore $\angle WXY=\angle ZYX$ .