Question

find the value of each variable and the measure of each angle.

Explanation:

Step1: Use vertical - angle property

Vertical angles are equal. So, \(5x - 52=x + 12\).

Step2: Solve for \(x\)

Subtract \(x\) from both sides: \(5x-x-52=x - x+12\), which gives \(4x-52 = 12\). Then add 52 to both sides: \(4x-52 + 52=12 + 52\), so \(4x=64\). Divide both sides by 4: \(x=\frac{64}{4}=16\).

Step3: Use another vertical - angle property

\(6y-12=x + 12\). Since \(x = 16\), then \(6y-12=16 + 12\).

Step4: Solve for \(y\)

First simplify the right - hand side: \(6y-12=28\). Add 12 to both sides: \(6y-12 + 12=28+12\), so \(6y=40\). Then \(y=\frac{40}{6}=\frac{20}{3}\).

Step5: Find angle measures

For the angle \(x + 12\), substitute \(x = 16\), the measure is \(16+12 = 28^{\circ}\). For the angle \(5x-52\), substitute \(x = 16\), \(5\times16-52=80 - 52=28^{\circ}\). For the angle \(6y-12\), substitute \(y=\frac{20}{3}\), \(6\times\frac{20}{3}-12=40 - 12 = 28^{\circ}\).

Answer:

\(x = 16\), \(y=\frac{20}{3}\)