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In the given diagram for problem 15, we are tasked with determining the measure of \( \angle MNP \) and identifying the theorem that applies. Let's break down the problem based on the provided information:

### Given:
- \( m \angle MNO = 50^\circ \)
- \( MP \perp MN \) (i.e., \( MP \) is perpendicular to \( MN \), meaning \( \angle MPN = 90^\circ \))
- \( OP \perp ON \) (i.e., \( OP \) is perpendicular to \( ON \), meaning \( \angle OPN = 90^\circ \))
- \( MP = OP \) (equal lengths for these segments)

### Objective:
- Find the measure of \( \angle MNP \).
- Identify the theorem used.

### Analysis:
1. Since \( MP = OP \) and both \( MP \perp MN \) and \( OP \perp ON \), triangle \( MOP \) is isosceles and both perpendicular segments create two right angles at points \( M \) and \( O \). 
2. In an isosceles triangle, the angles opposite the equal sides are also equal. This implies \( \angle MNP = \angle ONP \).
3. Given \( m \angle MNO = 50^\circ \), the remaining angle, which is \( \angle MNP \), must satisfy the properties of the triangle.

### Conclusion:
Based on the available options, and considering the equality and relationships between the angles and sides, the measure of \( \angle MNP \) is likely \( 25^\circ \). The theorem that applies is the **Angle Bisector Theorem**, which explains the equality in angles formed by a bisected angle in an isosceles triangle.

Would you like more details on how to apply the Angle Bisector Theorem or further clarifications? 

### 5 Related Questions:
1. What are the conditions for applying the Angle Bisector Theorem in triangles?
2. How do you solve for angles in isosceles triangles using perpendicular bisectors?
3. What are the properties of perpendicular lines in geometry?
4. Can the Angle Bisector Theorem be used in non-isosceles triangles?
5. How does symmetry in geometry help in solving angle problems?

### Tip:
When working with isosceles triangles, remember that the angle bisector always divides the opposite angle into two equal parts, making it easier to find unknown angles.

Given: m∠MNO = 50°, MP ⊥ MN, OP ⊥ ON, MP = OP. What is the measure of ∠MNP? By which Theorem?

Learn how to find the measure of ∠MNP in an isosceles triangle where MP = OP and both MP ⊥ MN and OP ⊥ ON. This problem utilizes the Angle Bisector Theorem and covers key geometric concepts. Suitable for students in Grades 9-11.

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