Question

the figure is not drawn to scale.

1. if m∠acd = 120 - x, m∠b = 60, and m∠a = 10 + x, find m∠acb.

85
35
25
50

Explanation:

Step1: Use exterior - angle property

The exterior - angle of a triangle is equal to the sum of the two non - adjacent interior angles. So, \(m\angle ACD=m\angle A + m\angle B\).
Substitute the given angle measures: \(120 - x=(10 + x)+60\).

Step2: Solve for \(x\)

First, simplify the right - hand side of the equation: \(120 - x=70 + x\).
Add \(x\) to both sides: \(120=70 + 2x\).
Subtract 70 from both sides: \(2x = 120 - 70=50\).
Divide both sides by 2: \(x = 25\).

Step3: Find \(m\angle ACB\)

Since \(\angle ACD\) and \(\angle ACB\) are supplementary (i.e., \(m\angle ACD+m\angle ACB = 180^{\circ}\)), and \(m\angle ACD=120 - x\).
Substitute \(x = 25\) into \(m\angle ACD\): \(m\angle ACD=120-25 = 95^{\circ}\).
Then \(m\angle ACB=180 - m\angle ACD\).
So \(m\angle ACB=180 - 95=85^{\circ}\).

Answer:

85