Question

1. if \\( \triangle a b c \\) is an isosceles triangle and \\( \triangle a d b e \\) is an equilateral triangle, find each angle measure
2. 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.

Explanation:

Step1: Use the angle - sum property of a triangle

In \(\triangle ABC\), the sum of the interior angles is \(180^{\circ}\). So, \(m\angle A + m\angle B+m\angle C=180^{\circ}\).
Substitute \(m\angle A=(3x + 18)^{\circ}\), \(m\angle B=(7x-58)^{\circ}\), and \(m\angle C=(2x - 8)^{\circ}\) into the equation:
\((3x + 18)+(7x-58)+(2x - 8)=180\)

Step2: Simplify the left - hand side of the equation

Combine like terms:
\(3x+7x + 2x+18-58 - 8=180\)
\(12x-48 = 180\)

Step3: Solve for \(x\)

Add \(48\) to both sides of the equation:
\(12x-48 + 48=180 + 48\)
\(12x=228\)
Divide both sides by \(12\):
\(x=\frac{228}{12}=19\)

Step4: Find the measure of each angle

• \(m\angle A=(3x + 18)^{\circ}\). Substitute \(x = 19\): \(m\angle A=3\times19+18=57 + 18=75^{\circ}\)
• \(m\angle B=(7x-58)^{\circ}\). Substitute \(x = 19\): \(m\angle B=7\times19-58=133-58 = 75^{\circ}\)
• \(m\angle C=(2x - 8)^{\circ}\). Substitute \(x = 19\): \(m\angle C=2\times19-8=38 - 8=30^{\circ}\)

Answer:

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