Question

Exterior Angle=72 degrees.

Find the interior angle measure
and the number of sides of the polygon.

Answer 1

The interior angle would measure
`108^(\circ)`. Assuming that this polygon is regular, it would contain
`5` sides.

Explanation:

An exterior angle in a polygon is supplementary with the interior angle that shares the same vertex with the exterior angle. In other words, the sum of these two angles would be
`180^(\circ)`.

In this question, the exterior angle measures
`72^(\circ)`. Therefore, the interior angle that shares the same vertex with this
`72^(\circ)\!` exterior angle would measure
`(180^(\circ) - 72^(\circ))`, which is
`108^(\circ)`.

The sum of all interior angles in a polygon with
`n` sides (regular or not) is
`180\, (n - 2)` degrees.

All the interior angles in a regular polygon are equal. Hence, in a regular polygon with
`n` sides (and hence
`n\!` vertices,) each of the
`n\!\!` interior angles would measure
`180\, (n - 2) / n` degrees.

Assume that the polygon in this question is regular. Again, let
`n` be the number of sides in this polygon. Each interior angle would measure
`180\, (n - 2) / n` degrees. However, it was also deduced that an interior angle of this polygon measures
`108^(\circ)`. That is:

`\displaystyle (180 \, (n - 2))/(n) = 108`.

Solve for
`n`:

`180\, n - 2 * 180 = 108\, n`.

`(180 - 108)\, n = 360`.

`\begin{aligned}n &= (360)/(180 -108) \\ &= (360)/(72) \\ &= 5\end{aligned}`.

In other words, if this polygon is regular, it would contain
`5` sides.