Question

in the diagram below of triangle nop, q is the mid - point of $overline{np}$ and r is the mid - point of $overline{op}$. if m$angle$pon = 45 + 3x, and m$angle$prq = 9 + 7x, what is the measure of $angle$prq?

Explanation:

Step1: Use mid - point theorem

Since $Q$ is the mid - point of $\overline{NP}$ and $R$ is the mid - point of $\overline{OP}$, by the mid - point theorem, $RQ\parallel ON$.

Step2: Identify corresponding angles

When $RQ\parallel ON$, $\angle PRQ$ and $\angle PON$ are corresponding angles, so $\angle PRQ=\angle PON$.

Step3: Set up the equation

Set $45 + 3x=9 + 7x$.

Step4: Solve the equation for $x$

Subtract $3x$ from both sides: $45=9 + 7x-3x$, which simplifies to $45=9 + 4x$.
Subtract $9$ from both sides: $45 - 9=4x$, so $36 = 4x$.
Divide both sides by $4$: $x = 9$.

Step5: Find the measure of $\angle PRQ$

Substitute $x = 9$ into the expression for $\angle PRQ$: $m\angle PRQ=9+7x$.
$m\angle PRQ=9+7\times9=9 + 63=72$.

Answer:

$72$