Question

1. find the values of x and y.

Explanation:

Step1: Set up equation for vertical - angles

Vertical - angles are equal. So, \(10x - 53=2x + 5\).

Step2: Solve for \(x\)

Subtract \(2x\) from both sides: \(10x-2x - 53=2x-2x + 5\), which simplifies to \(8x-53 = 5\). Then add 53 to both sides: \(8x-53 + 53=5 + 53\), giving \(8x=58\). Divide both sides by 8: \(x=\frac{58}{8}=\frac{29}{4}=7.25\).

Step3: Set up equation for linear - pair

The angles \((10x - 53)\) and \((18y + 11)\) form a linear - pair, so \((10x - 53)+(18y + 11)=180\). Substitute \(x = 7.25\) into the equation: \((10\times7.25-53)+(18y + 11)=180\). First, calculate \(10\times7.25-53=72.5 - 53 = 19.5\). The equation becomes \(19.5+18y + 11=180\). Combine like - terms: \(18y+30.5 = 180\). Subtract 30.5 from both sides: \(18y=180 - 30.5=149.5\). Divide both sides by 18: \(y=\frac{149.5}{18}=\frac{299}{36}\approx8.31\).

Answer:

\(x = 7.25\), \(y=\frac{299}{36}\approx8.31\)