Question

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

Explanation:

Step1: Set up equation for vertical - angles

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

Step2: Solve for \(x\)

Combining like - terms gives \(4x = 64\), then \(x=\frac{64}{4}=16\).

Step3: Set up equation for other vertical - angles

We also have \(6y-12\) and \(x + 12\). Since \(x = 16\), then \(x + 12=28\). So, \(6y-12 = 28\).
\[6y=28 + 12\]

Step4: Solve for \(y\)

\[6y=40\], then \(y=\frac{40}{6}=\frac{20}{3}\).

Step5: Find angle measures

One angle is \(x + 12\), substituting \(x = 16\), we get \(16+12 = 28^{\circ}\). Another angle is \(5x - 52\), substituting \(x = 16\), we get \(5\times16-52=80 - 52 = 28^{\circ}\). The angle \(6y-12\) is also \(28^{\circ}\) (because of vertical - angle equality), and the fourth angle is \(180 - 28=152^{\circ}\) (since adjacent angles are supplementary).

Answer:

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