15. 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 the equation 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), getting (9x=71), and (x=\frac{71}{9}\approx7.89).

Step3: Use the fact that adjacent angles are supplementary

Let's assume the adjacent - angle relationship. If we consider the linear - pair or other angle - sum property (assuming the angles are part of a linear pair or a known angle - sum situation). But if we assume the two non - vertical angles are supplementary (a common case in angle problems with intersecting lines), we have no information about the third angle related to ((18y + 5)) to form an equation. However, if we assume the two non - vertical angles are vertical to each other, then (18y+5=x + 10). Substitute (x=\frac{71}{9}) into the equation: (18y+5=\frac{71}{9}+10). First, simplify the right - hand side: (\frac{71}{9}+10=\frac{71 + 90}{9}=\frac{161}{9}). Then, (18y=\frac{161}{9}-5=\frac{161-45}{9}=\frac{116}{9}), and (y=\frac{116}{9\times18}=\frac{58}{81}\approx0.72).

Answer:

(x=\frac{71}{9},y = \frac{58}{81})