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Let's analyze the given geometry problem.

### Problem Details:
- We have an isosceles triangle \( \triangle MPN \) where \( MP \cong PN \).
- \( PR \parallel MN \), and the measure of angle \( \angle MPN = 86^\circ \).
- We are asked to find the measure of angle \( \angle 3 \).

### Approach:

1. **Isosceles Triangle \( \triangle MPN \)**: 
   Since \( MP \cong PN \), the base angles \( \angle 1 \) and \( \angle 2 \) are congruent.

2. **Sum of Angles in a Triangle**: 
   In any triangle, the sum of the interior angles is \( 180^\circ \).
   \[
   \angle MPN + \angle 1 + \angle 2 = 180^\circ
   \]
   Since \( \angle 1 = \angle 2 \), we have:
   \[
   86^\circ + 2\angle 1 = 180^\circ
   \]
   Solving for \( \angle 1 \):
   \[
   2\angle 1 = 180^\circ - 86^\circ = 94^\circ
   \]
   \[
   \angle 1 = \frac{94^\circ}{2} = 47^\circ
   \]
   So, \( \angle 1 = \angle 2 = 47^\circ \).

3. **Parallel Lines \( PR \parallel MN \)**:
   Since \( PR \parallel MN \), angle \( \angle 3 \) corresponds to angle \( \angle 1 \) because they are alternate interior angles.

### Conclusion:
Since \( \angle 1 = 47^\circ \), it follows that \( \angle 3 = 47^\circ \) as well (due to the alternate interior angles property).

Thus, the measure of \( \angle 3 \) is \( \boxed{47^\circ} \).

---

Would you like a more detailed explanation or have any other questions? 

Here are 5 related questions to explore:

1. What is the sum of the angles in any triangle?
2. How do you prove that alternate interior angles are congruent when lines are parallel?
3. In an isosceles triangle, why are the base angles always congruent?
4. What are other types of angle relationships formed by parallel lines and transversals?
5. Can you find the value of other unknown angles in a quadrilateral?

**Tip**: Always check if any lines in a diagram are parallel, as they often give useful angle relationships through properties like alternate interior angles.

Given that ΔMPN is isosceles with MP ≅ PN and PR || MN, if m∠MPN = 86°, find the measure of ∠3.

This problem involves finding the measure of angle 3 in an isosceles triangle ΔMPN with given angle relations. It covers key geometric concepts such as the angle sum property, base angles, and alternate interior angles with parallel lines. Suitable for Grades 7-9.

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