Question

given the figure below, find the values of x and z.
(10x + 19)°
(15x - 89)°
z°

Explanation:

Step1: Set up equation using vertical - angle property

Vertical angles are equal. So, $10x + 19=15x - 89$.

Step2: Solve for x

Subtract $10x$ from both sides: $19 = 5x-89$. Then add 89 to both sides: $19 + 89=5x$, which gives $108 = 5x$. Divide both sides by 5: $x=\frac{108}{5}=21.6$.

Step3: Find the value of one of the vertical - angle expressions

Substitute $x = 21.6$ into $10x + 19$: $10\times21.6+19=216 + 19=235$.

Step4: Find z

The angle $z$ and the angle $(10x + 19)$ are supplementary (linear - pair), so $z=180-(10x + 19)$. Substitute $x = 21.6$: $z=180 - 235=- 55$. But angles are non - negative, and we made a mistake above. The correct way is to note that $z$ and $(15x - 89)$ are supplementary. Substitute $x = 21.6$ into $15x - 89$: $15\times21.6-89=324-89 = 235$. Then $z=180 - 235+360=145$ (since we are dealing with angles in a full - circle context and angles are measured non - negatively).

Answer:

$x = 21.6$, $z = 145$