Question

1. in \\( \triangle a b c \\), if \\( a c \cong 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 \(AC\cong CB\), then \(\angle A=\angle B\) (base angles of an isosceles triangle are equal). So \(3x + 18=7x-58\).

Step2: Solve the equation for \(x\)

Subtract \(3x\) from both sides: \(18 = 4x-58\).
Add \(58\) to both sides: \(4x=18 + 58=76\).
Divide both sides by \(4\): \(x = 19\).

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

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

Step4: Calculate the measure of \(\angle C\)

Use the triangle - angle sum theorem (\(m\angle A+m\angle B+m\angle C=180^{\circ}\)).
\(m\angle C=180-(m\angle A + m\angle B)=180-(75 + 75)=30^{\circ}\).

Answer:

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