Question

1. in \\( \triangle wxy \\), if \\( \overline { w x } \cong \overline { w y } \\), \\( m \angle w = ( x + 6 ) ^ { \circ } \\), \\( m \angle x = ( 5 x - 12 ) ^ { \circ } \\), and \\( m \angle y = ( 7 x - 48 ) ^ { \circ } \\), find \\( x \\) and the measure of each angle.

\\( x = \\)
\\( m \angle w = \\)
\\( m \angle x = \\)
\\( m \angle y = \\)

Explanation:

Step1: Use the property of isosceles triangle

Since \(\overline{WX}\cong\overline{WY}\), then \(\angle X\cong\angle Y\). So \(5x - 12=7x - 48\).

Step2: Solve the equation for \(x\)

\(5x-12 = 7x - 48\)
\(48 - 12=7x - 5x\)
\(36 = 2x\)
\(x = 18\)

Step3: Calculate the measure of each angle

For \(m\angle W=(x + 6)^{\circ}\), substitute \(x = 18\), we get \(m\angle W=(18 + 6)^{\circ}=24^{\circ}\)
For \(m\angle X=(5x - 12)^{\circ}\), substitute \(x = 18\), \(m\angle X=(5\times18-12)^{\circ}=(90 - 12)^{\circ}=78^{\circ}\)
For \(m\angle Y=(7x - 48)^{\circ}\), substitute \(x = 18\), \(m\angle Y=(7\times18-48)^{\circ}=(126 - 48)^{\circ}=78^{\circ}\)

Answer:

\(x = 18\)
\(m\angle W=24^{\circ}\)
\(m\angle X = 78^{\circ}\)
\(m\angle Y=78^{\circ}\)