Question

The angles $P,Q,R$ and $S$ of $PQRS$ are in the ratio $1:3:7:9$, find the measure of each angle.

• A. ${19}^{\circ },{57}^{\circ },{136}^{\circ },{171}^{\circ }$
• B. ${18}^{\circ },{54}^{\circ },{126}^{\circ },{162}^{\circ }$
• C. ${17}^{\circ },{66}^{\circ },{116}^{\circ },{152}^{\circ }$
• D. ${12}^{\circ },{36}^{\circ },{108}^{\circ },{120}^{\circ }$

Answer 1

The correct option is B ${18}^{\circ },{54}^{\circ },{126}^{\circ },{162}^{\circ }$

Given ratio of angles of quadrilateral $PQRS$ is $1:3:7:9$

Let the angles of quadrilateral $PQRS$ be $x,3x,7x,9x$, respectively.

We know, by angle sum property, the sum of angles of a quadrilateral is ${360}^o$.

$\Rightarrow \angle P+\angle Q+\angle R+\angle S={360}^o$

$\Rightarrow x+3x+7x+9x={360}^o$

$\Rightarrow 20x={360}^o$

$\Rightarrow x=\frac{{360}^o}{20}$

$\therefore x={18}^o$.

$\therefore \angle P=x={18}^o$,

$\angle Q=3x=3\times {18}^o={54}^o$,

$\angle R=7x=7\times {18}^o={126}^o$

and $\angle S=9x=9\times {18}^o={162}^o$.

Hence, option $B$ is correct.