Question

in the diagram below of triangle $rst$, $u$ is the mid - point of $overline{rt}$ and $v$ is the mid - point of $overline{st}$. if m$angle tsr=-5x + 93$, and m$angle tvu=82 + 6x$, what is the measure of $angle tsr$?

Explanation:

Step1: Use mid - point theorem property

Since $U$ is the mid - point of $\overline{RT}$ and $V$ is the mid - point of $\overline{ST}$, by the mid - point theorem, $UV\parallel RS$. So, $\angle TVU=\angle TSR$ (corresponding angles).

Step2: Set up the equation

Set $-5x + 93=82 + 6x$.

Step3: Solve for $x$

Add $5x$ to both sides: $93=82 + 6x+5x$, which simplifies to $93=82 + 11x$. Then subtract 82 from both sides: $11x=93 - 82=11$. Divide both sides by 11, we get $x = 1$.

Step4: Find the measure of $\angle TSR$

Substitute $x = 1$ into the expression for $\angle TSR$: $\text{m}\angle TSR=-5(1)+93=-5 + 93=38$.

Answer:

$38$