Question

The given figure shows a sqaure, a regular pentagon and a regular hexagon put together, which of the following statements is true?

• A. $\angle P={42}^{\circ },\angle Q={69}^{\circ },\angle R={69}^{\circ }$
• B. $\angle P={40}^{\circ },\angle Q={60}^{\circ },\angle R={80}^{\circ }$
• C. $\angle P={46}^{\circ },\angle Q={67}^{\circ },\angle R={67}^{\circ }$
• D. $△PQR$ is isosceles.

Answer 1

Answer: (A), (D)

The correct options are
A

$\angle P={42}^{\circ },\angle Q={69}^{\circ },\angle R={69}^{\circ }$

D

$△PQR$ is isosceles.

(i) Each interior angle of a regular pentagon is = ${108}^{\circ }$

Each interior angle of a regular hexagon is = ${120}^{\circ }$

Each interior angle of a square is = ${90}^{\circ }$

$\angle QPR={360}^{\circ }-({108}^{\circ }+{90}^{\circ }+{120}^{\circ })$

$={360}^{\circ }-{318}^{\circ }$

$={42}^{\circ }$

(ii) Both pentagon and hexagon share the side of a same square.

$\Rightarrow$ Sides of both pentagon and hexagon are equal

$\therefore △PQR$ is isosceles (PQ = PR)

(iii) $\therefore \angle PQR=\angle PRQ=\frac{{180}^{\circ }-{42}^{\circ }}{2}$

$={69}^{\circ }$