Question

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

Explanation:

Step1: Use vertical - angle property

Vertical angles are equal. So $y = 70$ as it is equal to the given $70^{\circ}$ angle.

Step2: Set up an equation

Since the sum of angles around a point is $360^{\circ}$, and considering the pairs of vertical angles, we have $2y+2(x + y+35)=360$. Substitute $y = 70$ into the equation: $2\times70+2(x + 70+35)=360$.

Step3: Simplify the equation

$140+2(x + 105)=360$. Expand to get $140 + 2x+210 = 360$. Combine like - terms: $2x+350 = 360$.

Step4: Solve for x

Subtract 350 from both sides: $2x=360 - 350=10$. Divide both sides by 2: $x = 5$. But this is wrong. Let's use another approach. Since vertical angles are equal, $7x=x + y+35$. Substitute $y = 70$ into it: $7x=x + 70+35$.

Step5: Rearrange and solve for x

$7x−x=105$, $6x = 105$, $x = 35$.

The angles:
One pair of vertical angles is $y = 70^{\circ}$.
The other pair: $x + y+35=35 + 70+35=140^{\circ}$ and $7x=7\times35 = 140^{\circ}$.

Answer:

$x = 35$, $y=70$