Question

find the value of each variable and the measure of each angle.
x = (square) y = (square)

Explanation:

Step1: Use vertical - angles property

Vertical angles are equal. So, \(4x - 37=x + 11\).

Step2: Solve for \(x\)

Subtract \(x\) from both sides: \(4x - x-37=x - x + 11\), which simplifies to \(3x-37 = 11\). Then add 37 to both sides: \(3x-37 + 37=11 + 37\), so \(3x=48\). Divide both sides by 3: \(x = 16\).

Step3: Use linear - pair property

The angles \((4x - 37)\) and \((5y-x - 13)\) are a linear - pair, so their sum is \(180^{\circ}\). Substitute \(x = 16\) into \(4x-37\), we get \(4\times16-37=64 - 37 = 27\). Then the equation for \(y\) is \(27+(5y-16 - 13)=180\). Simplify the left - hand side: \(27+5y-29 = 180\), which gives \(5y-2 = 180\). Add 2 to both sides: \(5y-2 + 2=180 + 2\), so \(5y=182\). Divide both sides by 5: \(y=\frac{182}{5}=36.4\).

Step4: Find angle measures

Angle \(4x - 37\) with \(x = 16\) is \(4\times16-37=27^{\circ}\). Angle \(x + 11\) with \(x = 16\) is \(16 + 11=27^{\circ}\). Angle \(5y-x - 13\) with \(x = 16\) and \(y = 36.4\) is \(5\times36.4-16 - 13=182-16 - 13 = 153^{\circ}\). The other angle in the linear - pair with \(5y-x - 13\) is also \(153^{\circ}\).

Answer:

\(x = 16\), \(y = 36.4\)