Find the number of sides in a polygon, given that:
i. 5 interior angle each of $172{}^\circ $
ii. Remaining all are $160{}^\circ .$
Solution
Interior angle + Exterior angle $=~180{}^\circ $
$172{}^\circ ~+$ Exterior angle $=~180{}^\circ $
Exterior angle $=~180{}^\circ ~-~172{}^\circ ~=~8{}^\circ $
Same as, Exterior angle $=~180{}^\circ ~-~160{}^\circ ~=~20{}^\circ $
Now,
When, interior angle $172{}^\circ ,$ exterior angle $8{}^\circ $ (5 of them)
When, interior angle $160{}^\circ ,$ exterior angle $20{}^\circ $ (let 'n' of them)
Total exterior angle be $360{}^\circ $
Therefore, $8{}^\circ \times ~5~+~20{}^\circ \times ~~n~=~360{}^\circ $
$n~=~16$
Therefore, number of sides$~=~16~+~5~=~21$