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, $13x + 47=10x + 56$.

Step2: Solve for x

Subtract $10x$ from both sides: $13x-10x + 47=10x-10x + 56$, which simplifies to $3x+47 = 56$. Then subtract 47 from both sides: $3x+47 - 47=56 - 47$, giving $3x=9$. Divide both sides by 3: $x=\frac{9}{3}=3$.

Step3: Find the value of one of the vertical - angles

Substitute $x = 3$ into $13x + 47$. We get $13\times3+47=39 + 47=86$.

Step4: Find z

Since $z$ and the angle $13x + 47$ are supplementary (a straight - line is $180^{\circ}$), $z=180-(13x + 47)$. Substituting $x = 3$, we have $z=180 - 86=94$.

Answer:

$x = 3$
$z = 94$