Parallelogram with line AB parallel to line DC. Line AB equals 2y + 8. Line DC equals 3y - 4. Angle A equals 3x + 25. Angle C equals 5x - 5. Find the values of x and y in the parallelogram.

In a parallelogram, opposite sides are equal in length and opposite angles are equal in measure. The given information involves two parallel sides and two angles, so we can apply these properties to find the values of \( x \) and \( y \).

### Step 1: Set up the equation for the sides of the parallelogram
We are given that \( \overline{AB} \parallel \overline{DC} \), meaning \( AB = DC \). This gives the equation:

\[
2y + 8 = 3y - 4
\]

### Step 2: Solve for \( y \)
Now solve the equation to find \( y \):

\[
2y + 8 = 3y - 4
\]

Subtract \( 2y \) from both sides:

\[
8 = y - 4
\]

Add 4 to both sides:

\[
y = 12
\]

### Step 3: Set up the equation for the angles
We know that opposite angles in a parallelogram are equal. Therefore, \( \angle A = \angle C \). We are given:

\[
\angle A = 3x + 25 \quad \text{and} \quad \angle C = 5x - 5
\]

Since \( \angle A = \angle C \), we set the expressions equal to each other:

\[
3x + 25 = 5x - 5
\]

### Step 4: Solve for \( x \)
Now solve this equation:

\[
3x + 25 = 5x - 5
\]

Subtract \( 3x \) from both sides:

\[
25 = 2x - 5
\]

Add 5 to both sides:

\[
30 = 2x
\]

Divide both sides by 2:

\[
x = 15
\]

### Final Answers:
- \( x = 15 \)
- \( y = 12 \)

Would you like further clarification or assistance with any part of the solution?

### Follow-up Questions:
1. How can we apply the properties of parallelograms to other geometric shapes?
2. What are the implications of the angles being equal in a parallelogram?
3. What other ways could we approach solving this type of problem?
4. How do you use algebraic expressions to represent geometric properties like angles and sides?
5. What are the properties of different types of quadrilaterals and how do they relate to parallelograms?

### Tip:
When working with geometric shapes like parallelograms, always remember to use their fundamental properties, such as equal opposite sides and equal opposite angles, to set up your equations.

Learn how to solve for the unknowns x and y in a parallelogram with parallel sides and equal opposite angles. This problem involves algebraic equations and the properties of parallelograms, specifically focusing on the equality of opposite sides and angles. Suitable for students in Grades 9-11.

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