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To find \( IJ \) in the parallelogram \( HIJK \), we will use the property that opposite sides of a parallelogram are equal in length. In this case, sides \( HI \) and \( JK \) are equal, as are sides \( IJ \) and \( HK \).

Given:
- \( HK = 3z + 13 \)
- \( IJ = 9z - 5 \)

Since \( IJ = HK \) (opposite sides of a parallelogram are equal), we can set up the equation:

\[
9z - 5 = 3z + 13
\]

Now, let's solve this equation for \( z \):

\[
9z - 3z = 13 + 5
\]

\[
6z = 18
\]

\[
z = 3
\]

Now that we have the value of \( z \), substitute it back into the expression for \( IJ \):

\[
IJ = 9z - 5 = 9(3) - 5 = 27 - 5 = 22
\]

Thus, the length of \( IJ \) is 22 units.

Would you like a detailed breakdown of the steps or have any other questions? Here are some related questions you might find interesting:

1. How do we prove that opposite sides of a parallelogram are equal?
2. How would this problem change if the figure was a rectangle instead of a parallelogram?
3. Can you solve for \( HK \) directly from the given information?
4. How would you solve for an unknown angle in a parallelogram using similar techniques?
5. What properties of parallelograms can be used to solve problems involving diagonals?

**Tip:** In any polygon, focusing on symmetry or congruent parts often simplifies the problem!

Find IJ in parallelogram HIJK. Given HK = 3z + 13 and IJ = 9z - 5, find the length of IJ.

Learn how to solve for IJ in parallelogram HIJK using the properties of parallelograms. The problem involves setting up and solving an algebraic equation based on the equality of opposite sides. Suitable for Grade 8-10 students.

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