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We are given that triangles \( \triangle LMN \cong \triangle PQR \), which means they are congruent. Using the properties of congruent triangles, we can find the missing measures because corresponding parts of congruent triangles are equal (CPCTC).

Here is a step-by-step solution:

1. **Find \( m\angle P \):**
   Since \( \triangle LMN \cong \triangle PQR \), corresponding angles are congruent. Therefore:
   \[
   m\angle P = m\angle N = 45^\circ
   \]

2. **Find side \( QR \):**
   Again, by congruence, corresponding sides are equal. So, \( QR \) corresponds to \( MN \), and hence:
   \[
   QR = MN = 11.6 \, \text{units}
   \]

3. **Find \( m\angle M \):**
   By congruence, \( m\angle M \) corresponds to \( m\angle Q \). Thus:
   \[
   m\angle M = m\angle Q = 105^\circ
   \]

4. **Find side \( LN \):**
   By congruence, side \( LN \) corresponds to side \( PR \), so:
   \[
   LN = PR = 8.5 \, \text{units}
   \]

### Summary:
- \( m\angle P = 45^\circ \)
- \( QR = 11.6 \, \text{units} \)
- \( m\angle M = 105^\circ \)
- \( LN = 8.5 \, \text{units} \)

Would you like more details on the concepts used, or do you have any further questions?

Here are 5 related questions for further exploration:
1. How do we prove that two triangles are congruent using different criteria (SAS, ASA, etc.)?
2. Can you explain the property of corresponding parts of congruent triangles (CPCTC)?
3. What are other methods to find missing side lengths if triangles are not congruent?
4. How do we find unknown angles in a triangle if not all angles are given?
5. What is the significance of using degrees in measuring angles?

**Tip:** In congruent triangles, all corresponding angles and sides are equal, so you can directly transfer the values from one triangle to the other.

Given ΔLMN ≅ ΔPQR, find m∠P, QR, m∠M, and LN.

Learn how to solve for the missing angles and sides in congruent triangles ΔLMN ≅ ΔPQR using properties of congruent triangles (CPCTC). This problem involves calculating angle measures and side lengths based on corresponding parts. Suitable for students in Grades 9-10.

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