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 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 sum of angles in a linear - pair is \(180^{\circ}\). Let's assume \((10x - 61)\) and \((18y + 5)\) are in a linear - pair. First, substitute \(x=\frac{71}{9}\) into \(10x-61\): \(10\times\frac{71}{9}-61=\frac{710}{9}-\frac{549}{9}=\frac{710 - 549}{9}=\frac{161}{9}\). Then, since \((10x - 61)+(18y + 5)=180\), substitute the value of \(10x - 61\): \(\frac{161}{9}+18y + 5=180\). Convert 5 to \(\frac{45}{9}\), so \(\frac{161}{9}+\frac{45}{9}+18y=180\), \(\frac{161 + 45}{9}+18y=180\), \(\frac{206}{9}+18y=180\). Subtract \(\frac{206}{9}\) from both sides: \(18y=180-\frac{206}{9}=\frac{1620}{9}-\frac{206}{9}=\frac{1620 - 206}{9}=\frac{1414}{9}\). Then \(y=\frac{1414}{9\times18}=\frac{707}{81}\).

Answer:

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