The image shows a trapezoid with labeled angles. Let's define the trapezoid and solve for the variables $x$ and $y$, given the angles in terms of these variables.

Observations:

• The top left angle is $2y^\circ$.
• The top right angle is $4x^\circ$.
• The bottom left angle is $(2x + 12)^\circ$.
• The bottom right angle is $(y + 6)^\circ$.

Key Property of Trapezoids:

In a trapezoid, consecutive angles along the non-parallel sides are supplementary (i.e., they sum to $180^\circ$).

Step 1: Set up equations based on supplementary angles.

1. Top left and bottom left angles (along one non-parallel side): $2y + (2x + 12) = 180$ Simplifying: 2y + 2x + 12 = 180 \quad \Rightarrow \quad 2y + 2x = 168 \quad \Rightarrow \quad y + x = 84 \tag{1}

2. Top right and bottom right angles (along the other non-parallel side): $4x + (y + 6) = 180$ Simplifying: 4x + y + 6 = 180 \quad \Rightarrow \quad 4x + y = 174 \tag{2}

Step 2: Solve the system of equations.

We now have two equations: $y + x = 84 \quad \text{(Equation 1)}$ $4x + y = 174 \quad \text{(Equation 2)}$

We can solve this system using substitution or elimination. Let me solve it.The solution to the system of equations is:

• $x = 30^\circ$
• $y = 54^\circ$

Thus, the values for the variables $x$ and $y$ are $30^\circ$ and $54^\circ$ respectively.

Would you like further clarification on any steps? Here are some related questions you might want to explore:

1. How can we verify these values by substituting them back into the original angles?
2. What are the specific angle measures for each corner of the trapezoid?
3. Can we apply the same logic for trapezoids with different angle expressions?
4. How does the property of supplementary angles help in solving trapezoids?
5. Can you explain why consecutive angles along non-parallel sides must sum to $180^\circ$?

Tip: Always check your solutions by plugging them back into the original expressions to verify their correctness.