Exercise 8 Page 401

We are given the following pattern of regular polygons.

We are asked to find the measure of each angle at the circled vertex.

Note that each angle at the circled vertex is one of the interior angles of a regular polygon. To find the desired measures, we can use the rule for the sum of the measures of the interior angles of a polygon.

Interior Angle Sum of a Polygon

The sum of the measures of the interior angles of a polygon is $(n-2)180,$ where $n$ represents the number of sides.

We will start by finding the sum of the interior angles of each polygon. In the diagram, the circled vertex is the common vertex of a square, a regular hexagon, and a regular $12\text{-}$gon, or a dodecagon. Therefore, we will substitute $4$, $6,$ and $12$ for $n$ in the expression for the sum to find the sum of the interior measures in each polygon.

Polygon	$n$	$(n-2)180^{\circ }$	Simplify
square	$4$	$(4-2)180^{\circ }$	$360^{\circ }$
regular hexagon	$6$	$(6-2)180^{\circ }$	$720^{\circ }$
regular $12\text{-}$gon	$12$	$(12-2)180^{\circ }$	$1800^{\circ }$

Now, recall that in a regular polygon, all the angles have the same measure. To find the measure of one interior angle of each regular polygon, we will divide the sum of the angles by the number of angles in the polygon. Notice that in regular polygons the number of sides and the number of interior angles are equal. This means that we need to divide the sum of the angles by $n.$ Let's do it!

Polygon	$n$	Sum of Angles	One Angle
square	$4$	$360^{\circ }$	$\frac{360^{\circ }}{4}=90^{\circ }$
regular hexagon	$6$	$720^{\circ }$	$\frac{720^{\circ }}{6}=120^{\circ }$
regular $12\text{-}$gon	$12$	$1800^{\circ }$	$\frac{1800^{\circ }}{12}=150^{\circ }$

We found the measure of each angle at the circled vertex.

Now we can calculate the sum of the angles that we got. The sum of the angles is $360^{\circ }.$