Chapter 3.5, Problem 22WE

(a)

To determine

To Find: The measure of $\angle R,\angle S\text{and }\angle T$ .

Answer to Problem 22WE

  $\angle R=50°$ , $\angle S=150°$ and $m\angle T=150°$ .

Explanation of Solution

Given Information:

In a pentagon PQRST , $m\angle P=60°,m\angle Q=130°,\angle S\text{ and }\angle T$ are three times as large as $\angle R$ .

Calculation:

A pentagon has 5 sides.

The sum of interior angles in a pentagon is:

  $\begin{array}{r}\left(n-2\right)\times 180°=\left(5-2\right)\times 180°\\ =540°\end{array}$

It is given that $\angle S\text{ and }\angle T$ are three times as large as $\angle R$ , thus $\angle S=3\angle R\text{ and }\angle T=3\angle R$ .

So,

  $\begin{array}{c}m\angle P+m\angle Q+m\angle R+m\angle S+m\angle T=540°\\ 60°+130°+\angle R+3\angle R+3\angle R=540°\\ 7\angle R=540°-190°\\ 7\angle R=350°\\ \angle R=\frac{350°}{7}\\ \angle R=50°\end{array}$

Now, $\angle S=3\angle R\text{ and }\angle T=3\angle R$ .

  $\begin{array}{l}\angle S=\angle T=3\angle R\\ \angle S=\angle T=3\times 50°\\ \angle S=\angle T=150°\end{array}$

Therefore, $m\angle R=50°$ , $m\angle S=150°$ and $m\angle T=150°$ .

(b)

To determine

To Find: The pair of sides of a pentagon PQRST that must be parallel.

Answer to Problem 22WE

  $\overline{PQ}$ must be parallel to $\overline{RS}$ .

Explanation of Solution

Given Information:

  $m\angle P=60°$ , $m\angle Q=130°$ , $m\angle R=50°$ , $m\angle S=150°$ and $m\angle T=150°$ .

Calculation;

If sum of two angles is 180 degree, then angles are supplementary angles.

The sum of angles Q and R is:

  $\begin{array}{c}m\angle Q+m\angle R=130°+50°\\ =180°\end{array}$

Since, sum of $\angle Q$ and $\angle R$ is 180 degree therefore these angles are supplementary angles.

Since, in a pentagon if two opposite angles are supplementary then, sides containing angles are parallel therefore in pentagon PQRST $\overline{PQ}$ must be parallel to $\overline{RS}$ .