Answer:
The interior angle would measure [tex]108^{\circ}[/tex]. Assuming that this polygon is regular, it would contain [tex]5[/tex] sides.
Step-by-step 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 [tex]180^{\circ}[/tex].
In this question, the exterior angle measures [tex]72^{\circ}[/tex]. Therefore, the interior angle that shares the same vertex with this [tex]72^{\circ}\![/tex] exterior angle would measure [tex](180^{\circ} - 72^{\circ})[/tex], which is [tex]108^{\circ}[/tex].
The sum of all interior angles in a polygon with [tex]n[/tex] sides (regular or not) is [tex]180\, (n - 2)[/tex] degrees.
All the interior angles in a regular polygon are equal. Hence, in a regular polygon with [tex]n[/tex] sides (and hence [tex]n\![/tex] vertices,) each of the [tex]n\!\![/tex] interior angles would measure [tex]180\, (n - 2) / n[/tex] degrees.
Assume that the polygon in this question is regular. Again, let [tex]n[/tex] be the number of sides in this polygon. Each interior angle would measure [tex]180\, (n - 2) / n[/tex] degrees. However, it was also deduced that an interior angle of this polygon measures [tex]108^{\circ}[/tex]. That is:
[tex]\displaystyle \frac{180 \, (n - 2)}{n} = 108[/tex].
Solve for [tex]n[/tex]:
[tex]180\, n - 2 \times 180 = 108\, n[/tex].
[tex](180 - 108)\, n = 360[/tex].
[tex]\begin{aligned}n &= \frac{360}{180 -108} \\ &= \frac{360}{72} \\ &= 5\end{aligned}[/tex].
In other words, if this polygon is regular, it would contain [tex]5[/tex] sides.
