Question

PQRS is a parallelogram in which $\angle QRS$ and $\angle PQR$ are in ratio 7 : 3. Find the sum of $\angle PQR$ and $\angle PRQ.$

• A. ${117}^o$
• B. ${147}^o$
• C. ${110}^o$
• D. ${97}^o$

Answer 1

The correct option is A ${117}^o$

It is given that the ratio of $\angle QRT$ to $\angle PQR$ is 7 : 3
So,
let us assume $\angle QRS=7x$ and $\angle PQR=3x.$

Since,
$\angle QRS$ and $\angle PQR$ are adjacent angles of the parallelogram, they are supplementary.

So,
$\Rightarrow \angle QRS+\angle PQR={180}^o$
$\Rightarrow 7x+3x={180}^o$
$\Rightarrow x={18}^o$

By property of parallelogram, opposite angles are equal.

So, $\angle QRS=\angle SPQ=7x={126}^o$ and $\angle PQR=\angle PSR=3x={54}^o$

The diagonal of a parallelogram divides it into two congruent triangles.
So, the diagonal PR divides $\angle QPS$ and $\angle QRS$ into two equal halves.

So,
$\angle PRQ=\angle PRS=126/2={63}^o$

Hence, $\angle PQR+\angle PRQ={54}^o+{63}^o={117}^o$