Question

In the given figures, $PQRS$ is a parallelogram .Find the value of $x,y$ and $z$.

Answer 1

In figure (i),

$\angle PAS+\angle SAR={180}^{\circ }$ . . . [Linear pair of angles]

$x={180}^{\circ }-{90}^{\circ }$

$x={90}^{\circ }$

In $△SAR,$

$y={180}^{\circ }-\left({90}^{\circ }+{40}^{\circ }\right)$ . . . [Angle sum property]

$y={50}^{\circ }$

In parallelogram $PQRS,$

$\angle PSR=2\times \angle ASR$

$\angle PSR={100}^{\circ }$

So $\angle PQR={100}^{\circ }$ [opposite angles]

And hence $\angle z={50}^{\circ }$

In figure (ii),

$x={110}^{\circ }$ [opposite angle of parallelogram are equal]

and $x+z={180}^{\circ }$ [alternate interior angles]

$\Rightarrow z={70}^{\circ }$

Also $y+z={180}^{\circ }$ [linear pair of angles]

$\Rightarrow y={110}^{\circ }$