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We are given a parallelogram \( ABCD \), and the goal is to find \( x + y \). Let's work through the steps based on the diagram and the properties of a parallelogram.

### Properties of a parallelogram:
1. Opposite angles are equal.
2. Adjacent angles are supplementary (i.e., their sum is \( 180^\circ \)).

From the diagram:
- Angle \( A = y \)
- Angle \( D = (4y - 50)^\circ \)
- Angle \( B = (2x - 30)^\circ \)

Using the property that adjacent angles in a parallelogram are supplementary:
\[
y + (4y - 50) = 180
\]
Solving this equation for \( y \):
\[
5y - 50 = 180
\]
\[
5y = 230
\]
\[
y = 46
\]

Now that we know \( y = 46 \), let's move on to finding \( x \). The opposite angles of a parallelogram are equal, so:
\[
(2x - 30) = (4y - 50)
\]
Substitute \( y = 46 \) into the equation:
\[
2x - 30 = 4(46) - 50
\]
\[
2x - 30 = 184 - 50
\]
\[
2x - 30 = 134
\]
\[
2x = 164
\]
\[
x = 82
\]

### Final answer:
The value of \( x + y \) is:
\[
x + y = 82 + 46 = 128
\]

Would you like further details on the solution?

Here are five related questions to expand your understanding:
1. How do we know that adjacent angles in a parallelogram are supplementary?
2. What are some other properties of parallelograms that might be useful in different problems?
3. Could you solve for \( z \) based on the given information in the diagram?
4. How would the calculation change if the shape was not a parallelogram but a trapezoid?
5. How do we prove that opposite angles in a parallelogram are equal?

**Tip**: In any quadrilateral, especially parallelograms, focus on properties like angle relationships and parallel sides for efficient problem solving.

ABCD is a parallelogram. Find the value of x + y.

Learn how to solve for x + y in a parallelogram using algebraic expressions. This problem involves applying the properties of parallelograms and solving linear equations step by step. Suitable for Grades 8-10.

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