Question

1. determine the values of x and y.

Explanation:

Step1: Use vertical - angle property

Vertical angles are equal. $8y - 102=2y + 6$.

Step2: Solve for y

Subtract $2y$ from both sides: $8y-2y - 102=2y-2y + 6$, which simplifies to $6y-102 = 6$. Then add 102 to both sides: $6y-102 + 102=6 + 102$, so $6y=108$. Divide both sides by 6: $y=\frac{108}{6}=18$.

Step3: Use linear - pair property

$3x+(8y - 102)=180$ (since they form a linear - pair). Substitute $y = 18$ into $8y-102$: $8\times18-102=144 - 102 = 42$. Then the equation becomes $3x+42 = 180$. Subtract 42 from both sides: $3x=180 - 42=138$. Divide both sides by 3: $x=\frac{138}{3}=46$.

Answer:

$x = 46,y = 18$