Question

In a cyclic quadrilateral ABCD, if $AB||CD$ and $\angle B={70}^{\circ },$ find the remaining angles.

Answer 1

In a cyclic quadrilateral ABCD, AB || CD and $\angle B={70}^{\circ }$

$\because$ ABCD is a cyclic quadrilateral,

$\therefore \angle B+\angle D={180}^{\circ }$

$\Rightarrow {70}^{\circ }+\angle D={180}^{\circ }$

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

$\because AB||CD$

$\Rightarrow \therefore \angle A+\angle D={180}^{\circ }$ (Sum of cointerior angles)

$\angle A+{110}^{\circ }={180}^{\circ }$

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

Similarly, $\angle B+\angle C={180}^{\circ }$

$\Rightarrow {70}^{\circ }+\angle C={180}^{\circ }$

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

$\therefore \angle A={70}^{\circ }$, $\angle C={110}^{\circ },\angle D={110}^{\circ }$