Question

1. find the values of x and y.

Explanation:

Step1: Use vertical - angle property

Vertical angles are equal. So, \(10x - 61=x + 10\).

Step2: Solve the equation for \(x\)

Subtract \(x\) from both sides: \(10x-x-61=x - x+10\), which simplifies to \(9x-61 = 10\).
Add \(61\) to both sides: \(9x-61 + 61=10 + 61\), so \(9x=71\), and \(x=\frac{71}{9}\approx7.89\).
Also, adjacent angles are supplementary. So, \((10x - 61)+(18y + 5)=180\).
Substitute \(x = \frac{71}{9}\) into the equation: \(10\times\frac{71}{9}-61+18y + 5=180\).
\(\frac{710}{9}-61 + 5+18y=180\).
\(\frac{710}{9}-56+18y=180\).
\(\frac{710-504}{9}+18y=180\).
\(\frac{206}{9}+18y=180\).
Subtract \(\frac{206}{9}\) from both sides: \(18y=180-\frac{206}{9}=\frac{1620 - 206}{9}=\frac{1414}{9}\).
\(y=\frac{1414}{9\times18}=\frac{707}{81}\approx8.73\).

Answer:

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