Question

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

Explanation:

Step1: Apply mid - point theorem

Since \(U\) is the mid - point of \(\overline{RT}\) and \(V\) is the mid - point of \(\overline{ST}\), by the mid - point theorem, \(UV\parallel SR\).

Step2: Use corresponding angles

When \(UV\parallel SR\), \(\angle TVU\) and \(\angle TSR\) are corresponding angles, so \(\angle TVU=\angle TSR\).

Step3: Set up the equation

We set \(-5x + 93=82 + 6x\).

Step4: Solve for \(x\)

Add \(5x\) to both sides: \(93=82 + 6x+5x\), which simplifies to \(93=82 + 11x\). Then subtract 82 from both sides: \(11x=93 - 82=11\), so \(x = 1\).

Step5: Find the measure of \(\angle TSR\)

Substitute \(x = 1\) into the expression for \(\angle TSR\): \(\text{m}\angle TSR=-5(1)+93=-5 + 93=33\).

Answer:

33