Question

In the given figure, $MNOP$ is a cyclic quadrilateral.
If $\angle P=4x+12$ and $\angle N=3x+28$, then find the value of $x$ and also the measures of $\angle P$ and $\angle N$.

Answer 1

Here, $MNOP$ is a cyclic quadrilateral.

Given, $\angle P=4x+12,\angle N=3x+28$.

We know, in a cyclic quadrilateral, the sum of opposite angles are supplementary, i.e. ${180}^o$.

Here, $\angle P$ and $\angle N$ are opposite angles.

Then, $\angle P+\angle N={180}^o$

$⟹$ $(4x+12)+(3x+28)={180}^o$

$⟹$ $4x+3x+12+28={180}^o$

$⟹$ $7x+40={180}^o$

$⟹$ $7x={180}^o-{40}^o$

$⟹$ $7x={140}^o$

$⟹$ $x={20}^o$.

Then, $\angle P=4x+12=4\left({20}^o\right)+12={80}^o+12={92}^o$

and $\angle N=3x+28=3\left({20}^o\right)+28={60}^o+28={88}^o$.