Question

find the measure of angle 3. 3x + 101 4x + 115

Explanation:

Step1: Identify angle - pair relationship

Angles \(3x + 101\) and \(4x+115\) are supplementary (linear - pair, sum to \(180^{\circ}\)). So, \((3x + 101)+(4x + 115)=180\).

Step2: Simplify the equation

Combine like - terms: \(3x+4x+101 + 115=180\), which gives \(7x+216 = 180\).

Step3: Solve for \(x\)

Subtract 216 from both sides: \(7x=180 - 216=-36\), then \(x =-\frac{36}{7}\).

Step4: Find the measure of angle \(3\)

Angle \(3\) and the angle \(3x + 101\) are vertical angles, so they are equal. First, find the value of \(3x+101\):
Substitute \(x =-\frac{36}{7}\) into \(3x + 101\). \(3\times(-\frac{36}{7})+101=\frac{-108}{7}+\frac{707}{7}=\frac{-108 + 707}{7}=\frac{599}{7}\approx85.57^{\circ}\). Since angle \(3\) and \(3x + 101\) are vertical angles, the measure of angle \(3\) is also \(\frac{599}{7}\approx85.57^{\circ}\).

Answer:

\(\frac{599}{7}\) (or approximately \(85.57^{\circ}\))