Question

The figure shows a regular pentagon, a regular hexagon and an equilateral triangle put together. The measure of $\angle PQR$ is

• A. ${52}^{\circ }$
• B. ${62}^{\circ }$
• C. ${72}^{\circ }$
• D. ${82}^{\circ }$

Answer 1

The correct option is C

${72}^{\circ }$

Each angle of an equilateral triangle is ${60}^{\circ }$.

For regular pentagon, n=5.

$\Rightarrow$ Each exterior angle $=\frac{360}{5}={72}^{\circ }$

$={72}^{\circ }$

Now, $e+i={360}^{\circ }$ (Linear pair)

$72+i={180}^{\circ }$

$\Rightarrow i={108}^{\circ }$

For regular hexagon, n=6.

Each exterior angle $=\frac{360}{6}={60}^{\circ }$.

Now, $e+i={180}^{\circ }$ (Linear pair)

$\Rightarrow 60+i={180}^{\circ }$

$\Rightarrow i={120}^{\circ }$

Now, ${60}^{\circ }+{108}^{\circ }+{120}^{\circ }+\angle PQR={360}^{\circ }$...(Angles around a point)

$\Rightarrow \angle PQR+{288}^{\circ }={360}^{\circ }$

$\Rightarrow \angle PQR={360}^{\circ }-{288}^{\circ }$

$={72}^{\circ }$