Question

find the value of each variable and the measure of each angle.
(simplify your answers.)
x =

y =

(3x - y)°
(x + y)°
4y°

Explanation:

Step1: Use vertical - angle property

Vertical angles are equal. So we have two equations:
Equation 1: \(x + y=4y\), which simplifies to \(x = 3y\) (by subtracting \(y\) from both sides).
Equation 2: \(x + y=3x - y\), which simplifies to \(2y=2x\) or \(x = y\) (by first adding \(y\) to both sides and then subtracting \(x\) from both sides). But this is incorrect. Let's use the fact that the sum of angles around a point of intersection of two lines is \(360^{\circ}\), and vertical - angles are equal.
We know that \((x + y)+(3x - y)+4y+(x + y)=360^{\circ}\).
Combining like - terms gives \(5x + 5y=360^{\circ}\), or \(x + y = 72^{\circ}\). Also, since vertical angles are equal, \(x + y=4y\).

Step2: Solve the system of equations

From \(x + y=4y\), we get \(x=3y\).
Substitute \(x = 3y\) into \(x + y=72^{\circ}\), we have \(3y+y=72^{\circ}\), \(4y=72^{\circ}\), so \(y = 18^{\circ}\).
Substitute \(y = 18^{\circ}\) into \(x = 3y\), we get \(x=54^{\circ}\).

Answer:

\(x = 54\), \(y = 18\)