Question

given the figure below, find the values of x and z. (12x - 63)° (6x + 9)°

Explanation:

Step1: Set up equation using vertical - angle property

Vertical angles are equal. So, $12x - 63=6x + 9$.

Step2: Solve for x

Subtract $6x$ from both sides: $12x-6x - 63=6x-6x + 9$, which simplifies to $6x-63 = 9$. Then add 63 to both sides: $6x-63 + 63=9 + 63$, giving $6x=72$. Divide both sides by 6: $x=\frac{72}{6}=12$.

Step3: Find the measure of one of the vertical angles

Substitute $x = 12$ into $6x + 9$. So, $6\times12+9=72 + 9=81$.

Step4: Find z

Since the angle with measure $z$ and the angle $6x + 9$ are supplementary (linear - pair, sum to 180°), then $z=180-(6x + 9)$. Substituting $x = 12$ gives $z=180 - 81=99$.

Answer:

$x = 12$
$z = 99$