Question

In the adjacent figure HOPE is a parallelogram. Find the angle measures x, y and z. State the geometrical truths you use to find them.

Answer 1

In the figure, HOPE is a parallelogram in which HG is produced and HP is the diagonal

$\angle EHP={40}^{\circ }and\angle POQ={70}^{\circ }$
But $\angle POQ+\angle POH={180}^{\circ }$ (Linear pair)
$\Rightarrow {70}^{\circ }+w={180}^{\circ }\Rightarrow w={180}^{\circ }-{70}^{\circ }={110}^{\circ }$
But $\angle E=\angle POE$
(Opposite angles of a parallelogram)
$\therefore ={110}^{\circ }$
$\because$ HE || OP and HP is its transversal.
$\therefore \angle EHP=\angle HPO$ (Alternate angles)
$\Rightarrow {40}^{\circ }=y\Rightarrow y={40}^{\circ }$
In $\Delta PHO$
Ext. $\angle POQ=\angle PHO+\angle HPO\Rightarrow {70}^{\circ }=z+y\Rightarrow {70}^{\circ }=z+{40}^{\circ }\Rightarrow z={70}^{\circ }-{40}^{\circ }={30}^{\circ }$
Hence $x={110}^{\circ },y={40}^{\circ },z={30}^{\circ }$