Chapter 9, Problem 26CR

a.

To determine

Tocalculate:The sum ofmeasure of polygons angles.

Answer to Problem 26CR

Thesum of measure of polygons angles is ${180}^{\circ }$ .

Explanation of Solution

Given information:

A polygons with angles A, B, C, D, and E.

Formula used:

The below theorem is used:

The exterior angle theorem states that the measure of an exterior angle of a triangle is greater than either of the measures of the remote interior angles. The exterior angle of a triangle is equal to the sum of the two opposite interior angles.

  

Calculation:

  

By exterior angle theorem, we get

  $\begin{array}{l}\angle x=\angle E+\angle B\\ \angle y=\angle A+\angle D\end{array}$

Sum of angles in a triangle is ${180}^{\circ }$ .

  $\begin{array}{l}\angle x+\angle y+\angle C={180}^{\circ }\\ \angle C={180}^{\circ }-\angle x-\angle y----Equation1\\ \angle E+\angle B=\angle x----Equation2\\ \angle A+\angle D=\angle y----Equation3\end{array}$

Adding Equation 1, Equation 2 and Equation 3, we get

  $\angle A+\angle B+\angle C+\angle D+\angle E={180}^{\circ }$

b.

To determine

To find: The sum of angles measured depends on equilateral or equiangular polygons.

Answer to Problem 26CR

No,sum of angles measured does not depends on equilateral or equiangular polygons.

Explanation of Solution

Given information:

A polygons with angles A, B, C, D, and E.

Formula used:

The below theorem is used:

The exterior angle theorem states that the measure of an exterior angle of a triangle is greater than either of the measures of the remote interior angles. The exterior angle of a triangle is equal to the sum of the two opposite interior angles.

  

Calculation:

  

By exterior angle theorem, we get

  $\begin{array}{l}\angle x=\angle E+\angle B\\ \angle y=\angle A+\angle D\end{array}$

Sum of angles in a triangle is ${180}^{\circ }$ .

  $\begin{array}{l}\angle x+\angle y+\angle C={180}^{\circ }\\ \angle C={180}^{\circ }-\angle x-\angle y----Equation1\\ \angle E+\angle B=\angle x----Equation2\\ \angle A+\angle D=\angle y----Equation3\end{array}$

Adding Equation 1, Equation 2 and Equation 3, we get

  $\angle A+\angle B+\angle C+\angle D+\angle E={180}^{\circ }$

Equilateral polygons are those whose all sides are equal.

Equiangular polygons are those whose all angles are equal.

We use exterior angle theorem to find the sum of angles.