Question

In the figure, a regular octagon and a regular pentagon are put together. Find the measure of $\angle ABC$ .

Answer 1

For regular pentagon, n=5.

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

$={72}^{\circ }$ (1 mark)

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

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

$\Rightarrow i={108}^{\circ }$ (0.5 mark)

For a regular octagon, n=8.

$\therefore$ Each exterior angle $=\frac{360}{8}={45}^{\circ }$ (1 mark)

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

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

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

$\Rightarrow i={135}^{\circ }$ (0.5 mark)

Now, ${135}^{\circ }+{108}^{\circ }+\angle ABC={360}^{\circ }$ (Angles around a point)

$\Rightarrow \angle ABC+{243}^{\circ }={360}^{\circ }$

$\Rightarrow \angle ABC={360}^{\circ }-{243}^{\circ }$

$={117}^{\circ }$ (1 mark)