Question

1. in \\( \triangle qrs \\), if \\( \overline{qr} \cong \overline{rs} \\), \\( m \angle q = ( 8 x - 17 ) ^ { \circ } \\), \\( m \angle r = ( 19 x + 4 ) ^ { \circ } \\), and \\( m \angle s = ( 5 x + 1 ) ^ { \circ } \\), find the value of \\( x \\) and the measure of each angle.

\\( x = \\)
\\( m \angle q = \\)
\\( m \angle r = \\)
\\( m \angle s = \\)

Explanation:

Step1: Use the property of isosceles trapezoid

In an isosceles trapezoid \(QR\cong RS\), \(\angle Q=\angle S\). So \(8x - 17=5x + 1\).

Step2: Solve the equation for \(x\)

Subtract \(5x\) from both sides: \(8x-5x - 17=5x-5x + 1\), \(3x-17 = 1\).
Add \(17\) to both sides: \(3x-17 + 17=1 + 17\), \(3x=18\).
Divide both sides by \(3\): \(x=\frac{18}{3}=6\).

Step3: Calculate \(\angle Q\)

Substitute \(x = 6\) into \(m\angle Q=(8x - 17)^{\circ}\), \(m\angle Q=(8\times6-17)^{\circ}=(48 - 17)^{\circ}=31^{\circ}\).

Step4: Calculate \(\angle R\)

Substitute \(x = 6\) into \(m\angle R=(19x + 4)^{\circ}\), \(m\angle R=(19\times6+4)^{\circ}=(114 + 4)^{\circ}=118^{\circ}\).

Step5: Calculate \(\angle S\)

Since \(\angle Q=\angle S\), \(m\angle S = 31^{\circ}\).

Answer:

\(x = 6\), \(m\angle Q=31^{\circ}\), \(m\angle R = 118^{\circ}\), \(m\angle S=31^{\circ}\)