Question

if \\( \triangle pqr \\) is isosceles, what is the measure of \\( \angle q \\)?

Explanation:

Step1: Use the property of isosceles triangle

In an isosceles triangle \( \triangle PQR\) with \(QR = PR\), the base - angles are equal. So, the angles opposite to the equal sides are equal. That is, the angle adjacent to \((9x - 40)^{\circ}\) and the angle adjacent to \((5x + 8)^{\circ}\) are equal.
The sum of angles in a triangle is \(180^{\circ}\). Let's first find the value of \(x\).
We know that \((9x - 40)+(5x + 8)+(5x + 8)=180\) (because the two non - \( \angle Q\) angles are equal).

Step2: Simplify the equation

Combine like terms:

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$$

Add \(24\) to both sides of the equation: \(19x-24 + 24=180+24\), so \(19x=204\). Then \(x = 12\).

Step3: Calculate the measure of \(\angle Q\)

Substitute \(x = 12\) into the expression for \(\angle Q\).
\(\angle Q=(9x - 40)^{\circ}\).

$$
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$$

Answer:

\(68^{\circ}\)