Question

explain why only three types of regular polygons tessellate the plane.
choose the correct answer below.
a. in order for a regular polygon to tessellate the plane, its interior angle measure must be a divisor of 360°. only three regular polygons have interior angle measures that divide 360°. those polygons are the equilateral triangle, square, and regular hexagon.
b. in order for a regular polygon to tessellate the plane, its exterior angle measure must be a divisor of 360°. only three regular polygons have exterior angles that divide 360°. those polygons are the equilateral triangle, regular hexagon, and regular nonagon.
c. in order for a regular polygon to tessellate the plane, its angle bisectors must intersect at 90° angles.
d. in order for a regular polygon to tessellate the plane, its number of sides must divide 9. therefore only the equilateral triangle, regular hexagon and regular nonagon can tessellate the plane.

Explanation:

For a regular polygon to tessellate the plane, the interior angle must be a divisor of \(360^{\circ}\). The interior angles of an equilateral triangle (\(60^{\circ}\)), square (\(90^{\circ}\)), and regular hexagon (\(120^{\circ}\)) all divide \(360^{\circ}\) (\(360\div60 = 6\), \(360\div90=4\), \(360\div120 = 3\)). Other regular polygons like non - agons (interior angle \(\frac{(9 - 2)\times180^{\circ}}{9}=140^{\circ}\), \(360\div140=\frac{18}{7}\), not an integer) do not have interior angles that divide \(360^{\circ}\) evenly.

Answer:

A. In order for a regular polygon to tessellate the plane, its interior angle measure must be a divisor of \(360^{\circ}\). Only three regular polygons have interior angle measures that divide \(360^{\circ}\). Those polygons are the equilateral triangle, square, and regular hexagon.