Question

In a cyclic quadrilateral $ABCD$, twice the measure of $\angle A$ is thrice the measure of $\angle C$, find the measure of $\angle C$.

• A. ${36}^o$
• B. ${72}^o$
• C. ${90}^o$
• D. ${108}^o$

Answer 1

The correct option is B ${72}^o$

The sum of the opposite angles of a cyclic quadrilateral is supplementary.

So, in cyclic quadrilateral $ABCD$

$\angle A+\angle C={180}^{\circ }$ .............(1).

It is given that

$2\angle A=3\angle C$

$⟹$ $\angle A=\frac{3}{2}\angle C$ ............ (2).

From (1) and (2),

$\frac{3}{2}\angle C+\angle C={180}^{\circ }$

$⟹$$\frac{5}{2}\angle C={180}^{\circ }$

$⟹$$\angle C=\frac{180\times 2}{5}={72}^{\circ }$.

Hence, option $B$ is correct.