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Based on the problem given, we have a triangle \( \triangle JKL \) with \( Q \) as the incenter, meaning \( Q \) is the point of intersection of the angle bisectors of the triangle. We need to determine three measurements: \( m\angle PJM \), \( NQ \), and \( m\angle PLQ \).

### Finding \( m\angle PJM \)
The key point here is recognizing that \( Q \) is the incenter and thus, each of \( \angle JQK \), \( \angle KQL \), and \( \angle LQJ \) is split into two equal parts by the angle bisectors.

1. First, we calculate the remaining angles of \( \triangle JKL \). We know \( m\angle J = 42^\circ \) and \( m\angle P = 56^\circ \). As the internal angles of a triangle sum up to \( 180^\circ \):
   
   \[
   m\angle K = 180^\circ - 42^\circ - 56^\circ = 82^\circ
   \]

2. Since \( KQ \) and \( LQ \) are angle bisectors, \( m\angle KQJ = \frac{1}{2} m\angle K = 41^\circ \) and \( m\angle LQJ = \frac{1}{2} m\angle L = 28^\circ \).

3. Thus, \( m\angle PJM \) is the sum of these half angles as \( Q \) is the intersection point of the angle bisectors:

   \[
   m\angle PJM = m\angle KQJ + m\angle LQJ = 41^\circ + 28^\circ = 69^\circ
   \]

### Finding \( NQ \)
To find \( NQ \), we need to know more about the lengths of \( KQ \) and \( JQ \) or the properties related to the segments. Without these or additional relationships between the elements in the triangle, it isn't straightforward to compute \( NQ \) directly. Further information or assumptions might be required.

### Finding \( m\angle PLQ \)
Using the same logic as for \( m\angle PJM \):

1. \( m\angle KQJ = 41^\circ \) and \( m\angle KQL = \frac{1}{2} m\angle K = 41^\circ \).

2. Thus, \( m\angle PLQ = m\angle KQL + m\angle LQJ = 41^\circ + 28^\circ = 69^\circ \).

### Summary
1. \( m\angle PJM = 69^\circ \)
2. \( NQ \) cannot be calculated with the given information.
3. \( m\angle PLQ = 69^\circ \)

Would you like further details or have any questions about the process? Here are some related topics to explore:
1. What is the incenter of a triangle and its properties?
2. How to use angle bisectors to find specific lengths in a triangle?
3. How does the angle bisector theorem apply in different scenarios?
4. Can we determine the radius of the inscribed circle given the sides of the triangle?
5. What are other notable centers of a triangle, like the centroid, orthocenter, and circumcenter?

**Tip:** When solving geometry problems, always consider drawing additional lines like medians, altitudes, or angle bisectors; these often reveal symmetrical properties and helpful relationships.

The angle bisectors of ΔJKL are JQ, KQ, and LQ. They meet at a single point Q. (In other words, Q is the incenter of ΔJKL.) Suppose MQ=12, LQ=17, m∠MKN=42°, and m∠P JQ=56°. Find the following measures. Note that the figure is not drawn to scale.

Explore how to find measures in triangle JKL using properties of its incenter and angle bisectors. This problem involves calculating angles and cannot determine lengths without additional information. Ideal for high school students learning about geometric constructions and properties.

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