Question

Question 154
In trapezium HARE, EP and RP are bisectors of $\angle E$ and $\angle R$, respectively. Find $\angle HAR$ and $\angle EHA$.

Answer 1

Given, $\angle PER={25}^{\circ }$ and $\angle PRE={30}^{\circ }$
Since EP and PR are angle bisectors of $\angle REH$ and $\angle ARE$ respectively,
$\angle PEH={25}^{\circ }$ and $\angle PRA={30}^{\circ }$

$\angle E+\angle H={180}^{\circ }and\angle R+\angle A={180}^{\circ }$ [HARE is a trapezium]

$\Rightarrow \angle PER+\angle PEH+\angle H={180}^{\circ }and$
$\angle ERP+\angle PRA+\angle RAH={180}^{\circ }$

$\Rightarrow {25}^{\circ }+{25}^{\circ }+\angle H={180}^{\circ }and$
${30}^{\circ }+{30}^{\circ }+\angle A={180}^{\circ }$

$\Rightarrow {50}^{\circ }+\angle H={180}^{\circ }and{60}^{\circ }+\angle A={180}^{\circ }\Rightarrow \angle H={130}^{\circ }and\angle A={120}^{\circ }$

$\therefore \angle EHA={130}^{\circ }and\angle HAR={120}^{\circ }$.