Question

In the figure , PQRS is a rhombus in which the diagonal PR is produced

$\angle SRT={152}^{\circ }$, find x, y and z.

Answer 1

In rhombus PQRS, Diagonal PR and SQ bisect each other at right angles and PR is produced to T such that $\angle SRT={152}^{\circ }$

But $\angle SRT+\angle SRP={180}^{\circ }$ (Linear pair)

$\Rightarrow {152}^{\circ }+\angle SRP={180}^{\circ }\Rightarrow \angle SRP={180}^{\circ }-{152}^{\circ }={28}^{\circ }\text{But}\angle SPR=\angle SRP(\because PRbisects\angle Pand\angle R)\Rightarrow z={28}^{\circ }y={90}^{\circ }(\because \text{Diagonals bisect each other at rght angles})\angle RPQ=z={28}^{\circ }\therefore In\Delta POQ,z+x={90}^{\circ }\Rightarrow {28}^{\circ }+x={90}^{\circ }\Rightarrow x={90}^{\circ }-{28}^{\circ }={62}^{\circ }\therefore x={62}^{\circ },y={90}^{\circ },z={28}^{\circ }$