Question

1. (6 points) given \\( \triangle a b c, m \angle a=(3 x+28)^{\circ}, m \angle b=(5 x+52)^{\circ} \\), and \\( m \angle c=(2 x-10)^{\circ} \\). determine the value of \\( x \\) and \\( m \angle c \\).

Explanation:

Step1: Use the triangle angle - sum theorem

The sum of the interior angles of a triangle is \(180^{\circ}\). So, \(m\angle A+m\angle B + m\angle C=180^{\circ}\).
Substitute the given angle expressions: \((3x + 28)+(5x + 52)+(2x-10)=180\).

Step2: Simplify the left - hand side of the equation

Combine like terms: \((3x+5x + 2x)+(28 + 52-10)=180\).
\(10x+70 = 180\).

Step3: Solve for \(x\)

Subtract 70 from both sides: \(10x=180 - 70\).
\(10x=110\).
Divide both sides by 10: \(x=\frac{110}{10}=11\).

Step4: Find \(m\angle C\)

Substitute \(x = 11\) into the expression for \(m\angle C\).
\(m\angle C=(2x-10)^{\circ}\), so \(m\angle C=(2\times11 - 10)^{\circ}\).
First, calculate \(2\times11=22\), then \(22-10 = 12\).

Answer:

\(x = 11\) and \(m\angle C=12^{\circ}\)