Question

The figure shows a regular pentagon, a regular hexagon and a regular octagon put together . The measure of $\angle ABC$ and $\angle ADE$ are

• A. ${132}^{\circ },{105}^{\circ }$
• B. ${105}^{\circ },{132}^{\circ }$
• C. ${100}^{\circ },{137}^{\circ }$
• D. ${110}^{\circ },{112}^{\circ }$

Answer 1

The correct option is B

${105}^{\circ },{132}^{\circ }$

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)

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

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

For a regular octagon, n=8.

$\therefore$ Each exterior angle $=\frac{360}{8}={45}^{\circ }$

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

$\Rightarrow i+{45}^{\circ }={180}^{\circ }$

$\Rightarrow i={180}^{\circ }-{45}^{\circ }$

$\Rightarrow i={135}^{\circ }$

Now, ${120}^{\circ }+{135}^{\circ }+\angle ABC={360}^{\circ }$ (Angles around a point)

$\Rightarrow {255}^{\circ }+\angle ABC={360}^{\circ }$

$\Rightarrow \angle ABC={360}^{\circ }-{253}^{\circ }$

$\Rightarrow \angle ABC={105}^{\circ }$

Also, ${108}^{\circ }+{120}^{\circ }+\angle ADE={360}^{\circ }$ (Angles around a point)

$\Rightarrow {228}^{\circ }+\angle ADE={360}^{\circ }$

$\Rightarrow \angle ADE={360}^{\circ }-{228}^{\circ }$

$\Rightarrow \angle ADE={132}^{\circ }$