Question

1. find the values of x and y.

(18y + 5)°
(10x - 61)°
(x + 10)°

Explanation:

Step1: Use vertical - angle property

Vertical angles are equal. So, $10x−61=x + 10$.

Step2: Solve for x

Subtract x from both sides: $10x−x−61=x−x + 10$, which simplifies to $9x−61 = 10$. Then add 61 to both sides: $9x−61+61=10 + 61$, so $9x=71$, and $x=\frac{71}{9}$.

Step3: Use linear - pair property

The angles $(18y + 5)$ and $(10x−61)$ form a linear - pair, so $(18y + 5)+(10x−61)=180$. Substitute $x = \frac{71}{9}$ into the equation: $18y+5+10\times\frac{71}{9}-61 = 180$. First, simplify $10\times\frac{71}{9}-61=\frac{710}{9}-\frac{549}{9}=\frac{710 - 549}{9}=\frac{161}{9}$. Then the equation becomes $18y+5+\frac{161}{9}=180$. Multiply through by 9 to clear the fraction: $162y+45 + 161=1620$. Combine like - terms: $162y+206=1620$. Subtract 206 from both sides: $162y=1620 - 206=1414$. So, $y=\frac{1414}{162}=\frac{707}{81}$.

Answer:

$x=\frac{71}{9},y=\frac{707}{81}$