Question

In figure, lines AB and CD intersects at O. If $\angle AOC+\angle BOE={70}^{\circ }$ and $\angle BOD={40}^{\circ }$, find $\angle BOE$ and reflex $\angle COE$.

Answer 1

$\text{Given}:\angle AOC+\angle BOE={70}^{\circ }\text{and}\angle BOD={40}^{\circ }$

Since AB and CD intersect at O,
$\angle AOC=\angle BOD={40}^{\circ }\left(\text{Vertically opposite angles}\right)$

$\text{Also},\angle AOC+\angle BOE={70}^{\circ }\left(\text{given}\right)$
$\Rightarrow {40}^{\circ }+\angle BOE={70}^{\circ }$
$\Rightarrow \angle BOE={70}^{\circ }-{40}^{\circ }$
$\Rightarrow \angle BOE={30}^{\circ }$

$\text{Now},\angle AOC+\angle COE+\angle BOE={180}^{\circ }$
${40}^{\circ }+\angle COE+{30}^{\circ }={180}^{\circ }$
$\angle COE={180}^{\circ }-{70}^{\circ }$
$\angle COE={110}^{\circ }$

$\text{Reflex}\angle COE={360}^{\circ }-\angle COE$
$\text{Reflex}\angle COE={360}^{\circ }-{110}^{\circ }={250}^{\circ }$

$\text{Hence, reflex}\angle COE={250}^{\circ }\text{and}\angle BOE={30}^{\circ }$