Let's solve the problems one by one.

Problem 7

You have two pairs of intersecting lines, forming vertical angles.

• The angles opposite each other are equal. Therefore, we can set up the following two equations:
  1. $9x + 12 = 3x$
  2. $4y - 10 = 3x$

Let's solve these two equations step by step.

Step 1: Solve for $x$ in the first equation: $9x + 12 = 3x$ Subtract $3x$ from both sides: $6x + 12 = 0$ Now subtract 12 from both sides: $6x = -12$ Finally, divide by 6: $x = -2$

Step 2: Use the value of $x = -2$ to solve for $y$ in the second equation: $4y - 10 = 3(-2)$ Simplify the right-hand side: $4y - 10 = -6$ Add 10 to both sides: $4y = 4$ Finally, divide by 4: $y = 1$

So the values of the variables are $x = -2$ and $y = 1$.

Problem 8

In this problem, you have parallel lines and a transversal. The angles involved are consecutive interior angles, meaning that their sum is $180^\circ$.

Set up the following equation for the sum of the two angles on the left side: $(5y - 4) + 3y = 180$ Simplify: $8y - 4 = 180$ Add 4 to both sides: $8y = 184$ Now divide by 8: $y = 23$

Next, set up the equation for the sum of the angles on the right side: $3y + (2x + 13) = 180$ Substitute $y = 23$: $3(23) + (2x + 13) = 180$ Simplify: $69 + 2x + 13 = 180$ Combine like terms: $2x + 82 = 180$ Subtract 82 from both sides: $2x = 98$ Finally, divide by 2: $x = 49$

So the values of the variables are $x = 49$ and $y = 23$.

Would you like more details or have any other questions?

Here are 5 follow-up questions:

1. Can you solve another pair of equations involving vertical angles?
2. How do consecutive interior angles behave in parallel lines?
3. Why is it important to recognize angle relationships in geometric problems?
4. Can you derive a formula to solve for variables in other angle pairs?
5. How would the results change if the lines were not parallel in problem 8?

Tip: Always remember that vertical angles are congruent (equal in measure), while consecutive interior angles on parallel lines sum to $180^\circ$.