Question

In the figure, sides $PQ$ and $RQ$ of $△PQR$ are produced to points $S$ and $T,$ respectively. If $\angle SPR={135}^{\circ }$ and $\angle PQT={110}^{\circ }$, find $2\angle PRQ.$

• A. ${90}^{\circ }$
• B. ${150}^{\circ }$
• C. ${110}^{\circ }$
• D. ${130}^{\circ }$

Answer 1

The correct option is D

${130}^{\circ }$

As,
${110}^{\circ }+\angle 2={180}^{\circ }$ [linear pair]
$\angle 2={180}^{\circ }–{110}^{\circ }\angle 2={70}^{\circ }$

Also,
$\angle 1+{135}^{\circ }={180}^{\circ }$ [linear pair]
$\angle 1={180}^{\circ }–{135}^{\circ }\angle 1={45}^{\circ }$

In $△PQR,\angle 1+\angle 2+\angle PRQ={180}^{\circ }$
[Angle sum property]
$\Rightarrow {45}^{\circ }+{70}^{\circ }+\angle PRQ={180}^{\circ }\Rightarrow \angle PRQ={180}^{\circ }–{115}^{\circ }\Rightarrow \angle PRQ={65}^{\circ }$

Therefore,
$2\angle PRQ=2\times {65}^{\circ }={130}^{\circ }$