Question

given the figure below, find the values of x and z.

Explanation:

Step1: Set up equation using vertical - angle property

Vertical angles are equal. So, $7x + 85=14x+71$.

Step2: Solve for $x$

Subtract $7x$ from both sides: $85 = 7x+71$. Then subtract 71 from both sides: $7x=85 - 71=14$. Divide both sides by 7, we get $x = 2$.

Step3: Find the measure of one of the vertical angles

Substitute $x = 2$ into $7x + 85$. So, $7\times2+85=14 + 85=99$.

Step4: Find the value of $z$

The sum of angles around a point is $360^{\circ}$, and the two pairs of vertical angles are equal. Since one of the vertical - angle pairs is $99^{\circ}$, and $z$ is part of the linear - pair with the $99^{\circ}$ angle. So, $z=180 - 99 = 81$.

Answer:

$x = 2$, $z = 81$