Fill in the blanks.
Question:

Fill in the blanks.

(i) Each interior angle of a regular octagon is (………)°.

(ii) The sum of all interior angles of a regular hexagon is (………)°.

(iii) Each exterior angle of a regular polygon is 60°. This polygon is a ………

(iv) Each interior angle of a regular polygon is 108°. This polygon is a ………

(v) A pentagon has ……… diagonals.

Solution:

(i) Octagon has 8 sides.

$\therefore$ Interior angle $=\frac{180^{\circ} n-360^{\circ}}{n}$

Int erior angle $=\frac{\left(180^{\circ} \times 8\right)-360^{\circ}}{8} \quad=135^{\circ}$

(ii) Sum of the interior angles of a regular hexagon $=(6-2) \times 180^{\circ}=720^{\circ}$

(iii) Each exterior angle of a regular polygon is $60^{\circ}$.

$\therefore \frac{360}{60}=6$

Therefore, the given polygon is a hexagon.

(iv) If the interior angle is $108^{\circ}$, then the exterior angle will be $72^{\circ}$.   (interior and exterior angles are supplementary)

Sum of the exterior angles of a polygon is 360°.

Let there be sides of a polygon.

$72 n=360$

$n=\frac{360}{72}$

$n=5$

Since it has 5 sides, the polygon is a pentagon.

(v) A pentagon has 5 diagonals.

If $n$ is the number of $s$ ides, the number of diagonals $=\frac{n(n-3)}{2}$

$\frac{5(5-3)}{2}$

$=5$