Question

∠rst and ∠usw are straight angles and (overrightarrow{sv}) bisects ∠rst. if (mangle wst = 36^{circ}), find (mangle usv).

Explanation:

Step1: Recall angle - bisector property

Since $\overrightarrow{SV}$ bisects $\angle RST$, then $\angle RSV=\angle VST$.

Step2: Use the fact that $\angle USW$ and $\angle RST$ are straight - angles

A straight - angle measures $180^{\circ}$. So $\angle USW = \angle RST=180^{\circ}$.

Step3: Given $\angle WST = 36^{\circ}$

We know that $\angle RST=\angle RSV+\angle VST$ and $\angle RST = 180^{\circ}$. Also, since $\overrightarrow{SV}$ bisects $\angle RST$, let $\angle RSV=\angle VST = x$. Then $2x=180^{\circ}-\angle WST$.

Step4: Calculate $\angle VST$

Substitute $\angle WST = 36^{\circ}$ into the equation $2x=180 - 36$. So $2x = 144^{\circ}$, and $x=\angle VST = 72^{\circ}$.

Step5: Calculate $\angle USV$

$\angle USV=\angle USW-\angle WSV$. Since $\angle USW = 180^{\circ}$ and $\angle WSV=\angle WST+\angle VST=36^{\circ}+72^{\circ}=108^{\circ}$, then $\angle USV=180^{\circ}-108^{\circ}=72^{\circ}$.

Answer:

$72^{\circ}$