Question

1. if △abc is equilateral, find x.
2. if △rst is isosceles, find m∠s

Explanation:

Step1: Use the property of equilateral triangle

In an equilateral triangle, each angle is \(60^{\circ}\). So, \(8x - 44=60\).

Step2: Solve the equation for \(x\)

Add \(44\) to both sides: \(8x=60 + 44\), \(8x=104\).
Divide both sides by \(8\): \(x=\frac{104}{8}=13\).

Step1: Use the property of isosceles triangle

In an isosceles triangle \(\triangle RST\), assume \(\angle R=\angle T\) (if \(RS = TS\)). So, \(15x-31=9x + 11\).

Step2: Solve the equation for \(x\)

Subtract \(9x\) from both sides: \(15x-9x-31=9x-9x + 11\), \(6x-31 = 11\).
Add \(31\) to both sides: \(6x=11 + 31\), \(6x=42\).
Divide by \(6\): \(x = 7\).

Step3: Find the measure of \(\angle R\) and \(\angle T\)

Substitute \(x = 7\) into \(\angle R\): \(15\times7-31=105 - 31=74^{\circ}\), \(\angle T=9\times7+11=63 + 11=74^{\circ}\).

Step4: Use the angle - sum property of a triangle

Since the sum of angles in a triangle is \(180^{\circ}\), \(\angle S=180-(74 + 74)\).
\(\angle S=180 - 148=32^{\circ}\).

Answer:

\(x = 13\)

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