To solve this problem, we are working with a trapezoid, and the four labeled angles are based on the variables $x$ and $y$. Since these angles make up the interior angles of a quadrilateral, we know that the sum of the interior angles of any quadrilateral is 360 degrees.

The four angles provided are:

1. $2y^\circ$ (upper-left)
2. $4x^\circ$ (upper-right)
3. $(2x + 12)^\circ$ (lower-left)
4. $(y + 6)^\circ$ (lower-right)

Step 1: Set up the equation for the sum of angles

We know the sum of these angles is 360 degrees, so we can write:

$2y + 4x + (2x + 12) + (y + 6) = 360$

Step 2: Combine like terms

Combine all the terms involving $x$ and $y$:

$(4x + 2x) + (2y + y) + 12 + 6 = 360$

This simplifies to:

$6x + 3y + 18 = 360$

Step 3: Subtract 18 from both sides

$6x + 3y = 342$

Step 4: Simplify further by dividing by 3

$2x + y = 114$

This is Equation 1.

Step 5: Identify a second equation

Since the two angles on the same side of a trapezoid are supplementary (i.e., they sum to 180 degrees), we can form another equation based on this fact. Let’s use the two angles on the left side, $2y^\circ$ and $(2x + 12)^\circ$:

$2y + (2x + 12) = 180$

Step 6: Simplify the second equation

$2y + 2x + 12 = 180$

Subtract 12 from both sides:

$2y + 2x = 168$

Divide through by 2:

$y + x = 84$

This is Equation 2.

Step 7: Solve the system of equations

Now we solve the system of equations:

1. $2x + y = 114$
2. $x + y = 84$

Subtract Equation 2 from Equation 1:

$(2x + y) - (x + y) = 114 - 84$

Simplifying:

$x = 30$

Now substitute $x = 30$ into Equation 2:

$30 + y = 84$

Solving for $y$:

$y = 54$

Final Answer:

$x = 30, \quad y = 54$

Would you like more details or have any questions?

Follow-up Questions:

1. How do you check that the sum of the angles is correct?
2. Can you generalize this process for any quadrilateral?
3. What would the equations look like if the angles were different?
4. How would the process change if this were a different type of polygon?
5. What happens if two angles are supplementary but non-adjacent?

Tip:

When dealing with angle sums in polygons, remember that the sum of the interior angles of an $n$-sided polygon is $(n-2) \times 180^\circ$.