Question

find (x) and (z) where two angles are ((8x - 5)^{circ}) and ((13x - 65)^{circ}), and another angle is (z^{circ}). (x=) (z=)

Explanation:

Step1: Use vertical - angle property

Vertical angles are equal. So, $8x - 5=13x - 65$.

Step2: Solve for $x$

Subtract $8x$ from both sides: $- 5 = 13x-8x - 65$.
Simplify to get $-5 = 5x - 65$.
Add 65 to both sides: $-5 + 65=5x$, so $60 = 5x$.
Divide both sides by 5: $x=\frac{60}{5}=12$.

Step3: Find angle measure

First, find the measure of one of the vertical angles. Substitute $x = 12$ into $8x - 5$.
$8\times12-5=96 - 5=91^{\circ}$.

Step4: Find $z$

Since $z$ and the vertical - angle are supplementary (a linear pair), $z = 180-(8x - 5)$.
Substitute $x = 12$: $z=180 - 91 = 89$.

Answer:

$x = 12$
$z = 89$