Question

1. in \\( \triangle a b c \\), if \\( \overline{a c} \cong \overline{c b} \\), \\( m \angle a=(3 x+18)^{\circ} \\), \\( m \angle b=(7 x-58)^{\circ} \\), and \\( m \angle c=(2 x-8)^{\circ} \\), find the value of \\( x \\) and the measure of each angle.

\\( x= \\)
\\( m \angle a= \\)
\\( m \angle b= \\)
\\( m \angle c= \\)

Explanation:

Step1: Use the property of isosceles triangle

Since \(\overline{AC}\cong\overline{CB}\), then \(\angle A=\angle B\) (base - angles of an isosceles triangle are equal). So \(3x + 18=7x-58\).
Subtract \(3x\) from both sides: \(18 = 4x-58\).
Add \(58\) to both sides: \(4x=18 + 58=76\).
Divide both sides by \(4\): \(x = 19\).

Step2: Calculate the measure of each angle

• For \(\angle A\):

Substitute \(x = 19\) into \(m\angle A=(3x + 18)^{\circ}\).
\(m\angle A=(3\times19 + 18)^{\circ}=(57+18)^{\circ}=75^{\circ}\).

• For \(\angle B\):

Since \(\angle A=\angle B\), \(m\angle B = 75^{\circ}\).

• For \(\angle C\):

Substitute \(x = 19\) into \(m\angle C=(2x-8)^{\circ}\).
\(m\angle C=(2\times19-8)^{\circ}=(38 - 8)^{\circ}=30^{\circ}\).

Answer:

\(x = 19\), \(m\angle A=75^{\circ}\), \(m\angle B = 75^{\circ}\), \(m\angle C=30^{\circ}\)