Question

Quadrilateral $MRPN$ is cyclic. If $\angle R={(5x-13)}^o,\angle N={(4x+4)}^o$, find measures of $\angle R$ and $\angle N$.

Answer 1

Given, $\angle R=(5x-13{)}^o,\angle N=(4x+4{)}^o$.

We know, the sum of opposite angles of a cyclic quadrilateral being ${180}^{\circ }$.

Then,

$\angle R+\angle N={180}^o$

$⟹$ $(5x-13{)}^o+(4x+4{)}^o={180}^o$

$⟹$ $5x+4x-{13}^o+4^o={180}^o$

$⟹$ $9x-9^o={180}^o$

$⟹$ $9x=9^o+{180}^o$

$⟹$ $9x={189}^o$

$⟹$ $x={21}^o$.

Therefore,

$\angle R=(5x-13{)}^o=5({21}^o)-{13}^o={105}^o-{13}^o={92}^o$

and $\angle N=(4x+4{)}^o=4({21}^o)+4^o={84}^o+4^o={88}^o$.