Question

given the figure below, find the values of x and z. (9x + 89)° (8x + 57)°

Explanation:

Step1: Set up equation using vertical - angle property

Vertical angles are equal. So, $9x + 89=8x + 57$.

Step2: Solve for x

Subtract $8x$ from both sides: $9x-8x+89 = 8x - 8x+57$, which simplifies to $x+89 = 57$. Then subtract 89 from both sides: $x=57 - 89=-32$.

Step3: Find the measure of one of the angles

Substitute $x = - 32$ into $9x + 89$: $9(-32)+89=-288 + 89=-199$. This is incorrect. Let's use the fact that adjacent angles are supplementary. So, $(9x + 89)+(8x + 57)=180$.

Step4: Combine like - terms

$9x+8x+89 + 57=180$, which gives $17x+146 = 180$.

Step5: Solve for x

Subtract 146 from both sides: $17x=180 - 146 = 34$. Then divide by 17: $x = 2$.

Step6: Find z

Since $z$ and $9x + 89$ are vertical angles, substitute $x = 2$ into $9x+89$. $9(2)+89=18 + 89 = 107$. So $z = 107$.

Answer:

$x = 2$
$z = 107$