Question

1. if (svperp rt), (mangle rsu=(17x - 3)^{circ}), and (mangle ust=(6x - 1)^{circ}), find each missing measure.

Explanation:

Step1: Set up equation based on angle - addition

Since $\overline{SV}\perp\overline{RT}$, $\angle RSU$ and $\angle UST$ are supplementary, so $(17x - 3)+(6x - 1)=180$.

Step2: Combine like - terms

$17x+6x-3 - 1 = 180$, which simplifies to $23x-4 = 180$.

Step3: Solve for x

Add 4 to both sides: $23x=180 + 4=184$. Then divide both sides by 23, so $x=\frac{184}{23}=8$.

Step4: Find $m\angle RSU$

Substitute $x = 8$ into the expression for $m\angle RSU$: $m\angle RSU=17x-3=17\times8 - 3=136 - 3=133^{\circ}$.

Step5: Find $m\angle UST$

Substitute $x = 8$ into the expression for $m\angle UST$: $m\angle UST=6x-1=6\times8 - 1=48 - 1=47^{\circ}$.

Step6: Find $m\angle WSV$

Since $\overline{SV}\perp\overline{RT}$, $m\angle WSV = 90^{\circ}$.

Step7: Find $m\angle VSU$

$m\angle VSU=m\angle WSV + m\angle WSU$. Since $\angle WSV = 90^{\circ}$ and $\angle WSU$ is part of the linear - pair with $\angle RSU$, and we know $\angle RSU = 133^{\circ}$, $\angle WSU=180^{\circ}-\angle RSU = 47^{\circ}$, so $m\angle VSU=90^{\circ}+47^{\circ}=137^{\circ}$.

Answer:

$m\angle RSU = 133^{\circ}$, $m\angle UST = 47^{\circ}$, $m\angle WSV = 90^{\circ}$, $m\angle VSU = 137^{\circ}$