Question

1. find value of x
2. find the value of x and y. then find the measure of each angle.

6x° (15x + 75)° 3y°

Explanation:

Step1: Use vertical - angle property

Vertical angles are equal. So, $6x=15x + 75$.

Step2: Solve for x

Subtract $6x$ from both sides: $0 = 15x-6x + 75$, which simplifies to $0 = 9x+75$. Then subtract 75 from both sides: $-75 = 9x$. Divide both sides by 9: $x=-\frac{75}{9}=-\frac{25}{3}$.

Step3: Find the measure of angles

The first angle is $6x$. Substitute $x = -\frac{25}{3}$ into it: $6\times(-\frac{25}{3})=-50$. But angle measure cannot be negative in this context. We made a mistake above. The correct relationship is that $6x+(15x + 75)=180$ (since they are supplementary angles).

Step4: Solve the correct equation for x

Combine like - terms: $6x+15x+75 = 180$, so $21x+75 = 180$. Subtract 75 from both sides: $21x=180 - 75=105$. Divide both sides by 21: $x = 5$.

Step5: Find the measure of each angle

One angle is $6x$, substituting $x = 5$ gives $6\times5 = 30^{\circ}$. The other angle is $15x + 75$, substituting $x = 5$ gives $15\times5+75=75 + 75=150^{\circ}$.
Since $3y$ and $6x$ are vertical angles, $3y=6x$. Substitute $x = 5$ into it: $3y=6\times5$, so $y = 10$.

Answer:

$x = 5$, $y = 10$, the angles are $30^{\circ}$ and $150^{\circ}$