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To solve the problem in the diagram, we need to find the measure of \(\angle N\) in the isosceles triangle \( \Delta LMO \).

### Step-by-Step Solution:

1. **Given:**
   - \( \Delta LMO \) is isosceles with \( LO = MO \).
   - \( \angle L = 55^\circ \).
   - \( \angle NOM = 28^\circ \).

2. **Identify the angle \( \angle OLM \):**
   Since \( \Delta LMO \) is isosceles and \( LO = MO \), the base angles \( \angle LMO \) and \( \angle MLO \) are equal. We are given \( \angle L = 55^\circ \), so:
   \[
   \angle LMO = 55^\circ
   \]

3. **Calculate \( \angle OML \):**
   The angle \( \angle NOM \) is the exterior angle to \( \Delta LOM \). By the exterior angle theorem:
   \[
   \angle NOM = \angle LMO + \angle MLO
   \]
   But since \( \angle LMO = \angle MLO \), let's denote them as \( x \). So:
   \[
   \angle NOM = 2x
   \]
   And given \( \angle NOM = 28^\circ \):
   \[
   28^\circ = 2x
   \]
   \[
   x = 14^\circ
   \]

4. **Find \( \angle N \):**
   Now, using the fact that the sum of angles in \( \Delta LMN \) must be \( 180^\circ \), and we know \( \angle L = 55^\circ \) and \( \angle MNO = 28^\circ \), we can calculate \( \angle N \):
   \[
   \angle N = 180^\circ - \angle L - \angle NOM
   \]
   \[
   \angle N = 180^\circ - 55^\circ - 28^\circ
   \]
   \[
   \angle N = 97^\circ
   \]

### Final Answer:
The measure of \( \angle N \) is \( \boxed{97^\circ} \).

Do you want more details, or have any questions? Here are some related questions:

1. How would the problem change if \( LO \neq MO \)?
2. Can you find the lengths of the sides given more information?
3. How does the exterior angle theorem help in solving isosceles triangle problems?
4. What if \( \angle NOM \) was different? How would that affect \( \angle N \)?
5. How can you verify the sum of the angles in \( \Delta LMN \) using a different approach?

**Tip:** In isosceles triangles, the base angles (the angles opposite the equal sides) are always equal. This property can simplify solving for unknown angles.

In the diagram, ∆LMO is isosceles with LO = MO. If ∠L = 55° and ∠NOM = 28°, what is ∠N?

Learn how to find the angle in an isosceles triangle where one of the base angles is given as 55° and the angle at the vertex is given as 28°. This problem involves key geometry concepts such as the exterior angle theorem and properties of isosceles triangles. Suitable for students in Grades 7-9.

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