Question

part 1 of 2
find the value of each variable and the measure of each angle.
x = 16
y = 36.4

Explanation:

Step1: Use vertical - angle 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: $x=\frac{48}{3}=16$.

Step3: Use another angle - relationship

Since $4y-x - 11=x + 11$ (vertical - angle property or linear - pair relationship with equal vertical angles). Substitute $x = 16$ into the equation: $4y-16-11=16 + 11$.

Step4: Solve for $y$

First, simplify the left - hand side and right - hand side: $4y-27 = 27$. Add 27 to both sides: $4y-27 + 27=27 + 27$, so $4y=54$. Divide both sides by 4: $y=\frac{54}{4}=13.5$.

Step5: Find angle measures

For the angle $(x + 11)^{\circ}$, substitute $x = 16$: $(16 + 11)^{\circ}=27^{\circ}$.
For the angle $(4x-37)^{\circ}$, substitute $x = 16$: $4\times16-37=64 - 37=27^{\circ}$.
For the angle $(4y-x - 11)^{\circ}$, substitute $x = 16$ and $y = 13.5$: $4\times13.5-16-11=54-16 - 11=27^{\circ}$.

Answer:

$x = 16$, $y = 13.5$, angles are $27^{\circ}$ each.