Question

The opposite angles of a cyclic quadrilateral is in the ratio $2:3.$ Find the angles.

• A. ${36}^o$
• B. ${144}^o$
• C. ${72}^o$
• D. ${108}^o$

Answer 1

Answer: (C), (D)

${108}^o$

Consider the following cyclic quadrilateral:
Let, $\angle X=2x$ and $\angle Z=3x$ such that $\frac{\angle X}{\angle Z}=\frac{2x}{3x}=2:3$

We know that the sum of opposite angle of a cyclic quadrilateral is ${180}^o.$
$\Rightarrow \angle X+\angle Z={180}^o$
$\Rightarrow 2x+3x={180}^o$
$\Rightarrow 5x={180}^o$
$\Rightarrow x=\frac{{180}^o}{5}={36}^o$

$\therefore \angle X=2x=2\times {36}^o={72}^o$
And, $\angle Z=3x=3\times {36}^o={108}^o$

Hence, the required two angles are ${72}^o$ and ${108}^o.$