Question

The adjacent figure HOPE is a parallelogram. find the angle measures $x,y$ and $z$. State the properties you use to find them.

Answer 1

Since HOPE is a parallelogram, therefore, $HE\parallel OP$ and

$HO\parallel EP$

Now, $HE\parallel OP$ and transversal HO intersects them.

$\therefore \angle EHO=\angle POX[\because$ Corresponding angles are equal]

$\angle {40}^{\circ }+z={70}^{\circ }\Rightarrow z={70}^{\circ }-{40}^{\circ }={30}^{\circ }$

Again, $HE\parallel OP$ and transversal HP intersects them.

$\therefore \angle OPH=\angle EHP[\because$ Alternate angles are equal)

$\Rightarrow y={40}^{\circ }$

Since opposite angles are equal in a parallelogram.

$\therefore \angle HEP=\angle HOP$

$\Rightarrow x={180}^{\circ }-\angle POX={180}^{\circ }-{70}^{\circ }={110}^{\circ }$

Hence, $x={110}^{\circ },y={40}^{\circ }$ and $z={30}^{\circ }$