Question

$\overrightarrow{SW}$bisects $\angle RST$and $m\angle RST=72$. $\overrightarrow{SZ}$bisects $\angle RSW$, and $\overrightarrow{SR}$bisects $\angle NSW$. Sketch a diagram and find $m\angle RSZ$and$m\angle RSZ$

Short answer

The diagram is:

And

$m\angle RSZ={\mathbf{18}}^{\circ }$,$m\angle NSZ={\mathbf{54}}^{\circ }$

Step-by-step solution

Step 1. Sketch the diagram.

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

Therefore, the diagram is:

Step 2. Calculate m∠RSZ.

Since $\overrightarrow{SW}$bisects $\angle RST$,

$$\begin{gathered} \angle RSW=\angle TSW=\frac{1}{2}\angle RST \\ \angle RSW=\angle TSW=\frac{1}{2}\times {72}^{\circ } \\ \angle RSW=\angle TSW={36}^{\circ } \end{gathered}$$

Now, it is known that $\overrightarrow{SZ}$bisects $\angle RSW$, so,

$\angle RSZ=\angle WSZ=\frac{1}{2}\angle RSW$

Since $\angle RSW={36}^{\circ }$, then,

$$\begin{gathered} \angle RSZ=\angle WSZ=\frac{1}{2}\times {36}^{\circ } \\ \angle RSZ=\angle WSZ={18}^{\circ } \\ \angle RSZ={18}^{\circ } \end{gathered}$$

Step 3. Calculate m∠NSZ.

Again $\overrightarrow{SR}$bisects $\angle NSW$and $\angle RSW={36}^{\circ }$, so,

$$\begin{gathered} \angle NSR=\angle RSW=\frac{1}{2}\angle NSW \\ \angle NSR=\angle RSW={36}^{\circ } \\ \angle NSR={36}^{\circ } \end{gathered}$$

For $\angle NSZ$,

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

Step 4. State the conclusion.

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