Let us solve this step by step using the provided information.

Key Information:

1. $Z$ is the incenter of $\triangle PQR$, meaning it is the intersection of the angle bisectors.
2. Given:
   • $WZ = 14$
   • $QZ = 17$
   • $\angle WQX = 92^\circ$
   • $\angle YRZ = 16^\circ$.

The goal is to find specific measures using these inputs.

Step 1: Relationship between the angles

The incenter $Z$ has these properties:

1. It is equidistant from all sides of $\triangle PQR$, meaning the segments $WZ$, $YZ$, and $XZ$ are perpendicular distances from $Z$ to $\overline{PQ}$, $\overline{PR}$, and $\overline{QR}$, respectively.
2. $\angle WQX = 92^\circ$ tells us that the configuration at $Q$ forms a right angle at $WZ$, hinting that $WQ$ is perpendicular to $QZ$.
3. Similarly, $\angle YRZ = 16^\circ$ involves the angle formed at $Z$ along the bisector towards $R$, helping analyze its geometry.

Step 2: Perpendicular distances and triangle properties

• We are given $WZ = 14$ and $QZ = 17$.
• To calculate other measures, such as side lengths or angles in $\triangle PQR$, trigonometric relationships or coordinate geometry may be applied.

I'll proceed with detailed calculations next to extract exact measures. Let me compute them.To proceed further, let us determine all relevant measures systematically, based on the incenter's properties and relationships in $\triangle PQR$.

1. Use Incenter Properties

The incenter $Z$ satisfies:

• Perpendicular distances $WZ$, $YZ$, and $XZ$ from $Z$ to sides $\overline{PQ}$, $\overline{PR}$, and $\overline{QR}$, respectively, are the radii of the inscribed circle.

2. Use the Angle Information

Given:

• $\angle WQX = 92^\circ$
  • Suggests a near-right angle at $WQX$, helping confirm perpendicularity.
• $\angle YRZ = 16^\circ$
  • Involves the geometry of the triangle bisectors, which could assist in identifying $\angle PQR$, given the symmetry of angle bisectors.

Let me clarify a few additional details:

• Are we solving for side lengths ($\overline{PQ}$, $\overline{QR}$, $\overline{PR}$) or specific angles in $\triangle PQR$? This will ensure a focused approach.