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In the image, we are given a parallelogram \( LMNO \), with the angles at \( L \) and \( N \) defined algebraically. Specifically:

- \( \angle L = (25x)^\circ \)
- \( \angle N = (22x + 9)^\circ \)

The task is to determine the measure of \( \angle LMN \), assuming quadrilateral \( LMNO \) is a parallelogram.

### Step-by-Step Solution:

In a parallelogram, opposite angles are congruent, and adjacent angles are supplementary. That means:

1. \( \angle L \) and \( \angle N \) are supplementary, so their sum must be \( 180^\circ \):
   \[
   \angle L + \angle N = 180^\circ
   \]
   Substituting the expressions for \( \angle L \) and \( \angle N \):
   \[
   25x + (22x + 9) = 180
   \]
   
2. Simplify the equation:
   \[
   25x + 22x + 9 = 180
   \]
   \[
   47x + 9 = 180
   \]

3. Solve for \( x \):
   \[
   47x = 180 - 9
   \]
   \[
   47x = 171
   \]
   \[
   x = \frac{171}{47} \approx 3.64
   \]

4. Now that we have \( x \), substitute it back into the expression for \( \angle L \):
   \[
   \angle L = 25x = 25(3.64) \approx 91^\circ
   \]

Thus, \( \angle LMN = \boxed{91^\circ} \).

### Do you want further details or clarifications on any part?

Here are 5 related questions you might find useful:
1. What is the general rule for the sum of the interior angles of a quadrilateral?
2. How do we prove that opposite angles of a parallelogram are congruent?
3. How do we calculate the measures of all other angles in the parallelogram?
4. What happens if the quadrilateral is not a parallelogram but a trapezoid?
5. How do we handle algebraic expressions in geometric contexts?

**Tip:** In parallelograms, supplementary angles (those on the same side of a transversal) always add up to 180 degrees!

If quadrilateral LMNO is a parallelogram, what must the measure of angle LMN be?

Learn how to calculate the measure of angle LMN in parallelogram LMNO using algebraic expressions for the given angles. This solution involves using the properties of supplementary angles in parallelograms.

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