Chapter 2.3, Problem 14WE

To determine

To Find: the measure of angle $\angle RSZ\text{ and }\angle NSZ$ for the given conditions such as $\overrightarrow{SW}$ bisects $\angle RST$ , $m\angle RST=72$ , $\overrightarrow{SZ}$ bisects $\angle RSW$ and $\overrightarrow{SR}$ bisects $\angle NSW$ .

table for regular polygon for the measure of exterior and interior angles.

Answer to Problem 14WE

  $\angle RSZ=18°$ and $\angle NSZ=54°$

Explanation of Solution

Given:

  $\overrightarrow{SW}$ bisects $\angle RST,m\angle RST=72°$ , $\overrightarrow{SZ}$ bisects $\angle RSW$ and $\overrightarrow{SR}$ bisects $\angle NSW$ .

Sketch has been drawn as per given conditions.

  

  $\overrightarrow{SW}$ bisects $\angle RST$ .

  $\begin{array}{l}\angle RSW=\angle TSW=\frac{1}{2}\angle RST\\ \angle RSW=\angle TSW=\frac{1}{2}\times 72°\\ \angle RSW=\angle TSW=36°\end{array}$

  $\overrightarrow{SZ}$ bisects $\angle RSW$ .

  $\begin{array}{l}\angle RSZ=\angle WSZ=\frac{1}{2}\angle RSW\\ \therefore \left(\angle RSW=36\right)°\\ \angle RSZ=\angle WSZ=\frac{1}{2}\times 36°\\ \angle RSZ=18°\end{array}$

  $\overrightarrow{SR}$ bisects $\angle NSW$ .

  $\begin{array}{l}\angle NSR=\angle RSW=\frac{1}{2}\angle NSW\\ \therefore \left(\angle RSW=36°\right)\\ \angle NSR=\angle RSW=36°\\ \angle NSR=36°\end{array}$

For $\angle NSZ$ .

  $\begin{array}{l}\angle NSZ=\angle NSR+\angle RSZ\\ \angle NSZ=36°+18°\\ \angle NSZ={54}^{\circ }\end{array}$

Therefore, $\angle RSZ=18°$ and $\angle NSZ={54}^{\circ }$