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The problem states that \( \overrightarrow{NQ} \) bisects \( \angle MNP \), and we are given that \( m\angle QNM = 67^\circ \). Since \( \overrightarrow{NQ} \) is a bisector, it divides \( \angle MNP \) into two equal parts.

This means:

\[
m\angle MNP = 2 \times m\angle QNM
\]

Substituting the given value:

\[
m\angle MNP = 2 \times 67^\circ = 134^\circ
\]

Thus, \( m\angle MNP = 134^\circ \).

Would you like further details or explanations on this? Here are some follow-up questions you might find helpful:

1. What is an angle bisector, and how does it work?
2. How can you check if a line truly bisects an angle?
3. What happens if the bisector is not exact, and the two resulting angles are not equal?
4. Can angle bisectors be used in different geometric figures, like triangles and quadrilaterals?
5. How can the concept of angle bisectors be applied in coordinate geometry?

**Tip:** Always ensure that the given information matches the properties of geometric definitions, such as bisectors, to solve problems accurately.

If NQ bisects ∠MNP, and if m∠QNM = 67°, then find m∠MNP.

Learn how to calculate m∠MNP when NQ bisects the angle and m∠QNM = 67°. This problem uses the concept of angle bisectors, with detailed steps suitable for Grades 7-9.

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