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!