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Let's break down and solve this problem step by step.

### Given:
- \( m \angle X = 30^\circ \)
- \( m \angle Y \hat{T} = 50^\circ \)
- \( m \angle W \hat{R} \hat{Z} = 90^\circ \)
- \( \overline{XY} \) is tangent to the circle at point \( T \).

The problem asks to find the measure of angle \( Y \hat{Z} \).

### Analyzing the Diagram:
1. The line \( \overline{XY} \) is tangent to the circle at point \( T \), meaning \( \overline{XY} \) forms a right angle with the radius \( \overline{RT} \). Therefore, \( m \angle Y \hat{T} \hat{R} = 90^\circ \).

2. We are given that \( m \angle X = 30^\circ \). Also, \( m \angle Y \hat{T} = 50^\circ \).

3. \( m \angle W \hat{R} \hat{Z} = 90^\circ \) tells us that the line \( \overline{WRZ} \) is a straight line passing through the center of the circle.

### Applying the Tangent-Secant Theorem:
We can use the fact that the tangent \( \overline{XY} \) and the secant \( \overline{YZ} \) create a relationship with the angle at the point of tangency. The angle formed outside the circle by a tangent and a secant can be expressed as half the difference of the intercepted arcs.

Given the angles provided, we can use the geometry of circles to find the desired angle \( m \angle Y \hat{Z} \).

### Solution Process:
- We know that the sum of angles in a triangle is \( 180^\circ \). In triangle \( YTZ \), we have the following information:
  - \( m \angle Y \hat{T} = 50^\circ \)
  - \( m \angle T \hat{Z} = 90^\circ \) because \( \overline{XZ} \) is a diameter of the circle, so angle \( T \hat{Z} \hat{R} \) is a right angle.

Thus, we can find \( m \angle Y \hat{Z} \) using the formula for the sum of the angles in a triangle:

\[
m \angle Y \hat{Z} = 180^\circ - (m \angle Y \hat{T} + m \angle T \hat{Z})
\]
\[
m \angle Y \hat{Z} = 180^\circ - (50^\circ + 90^\circ)
\]
\[
m \angle Y \hat{Z} = 40^\circ
\]

Therefore, the measure of angle \( Y \hat{Z} \) is \( \boxed{40^\circ} \).

---

Would you like further details or clarifications on any part of the solution?

Here are some related questions to explore:
1. How is the tangent-secant theorem applied in different circle problems?
2. Why is angle \( T \hat{Z} \hat{R} \) a right angle in this specific case?
3. Can we solve this using any other geometric properties of circles?
4. How would the solution change if the tangent wasn't at point \( T \)?
5. What other properties of tangent lines are useful in geometry problems?

### Tip:
In problems involving tangents and secants, remember that angles formed outside the circle can often be calculated using the intercepted arcs formula.

Resuelve esto: Encuentra el ángulo YZ en el problema geométrico proporcionado en la imagen.

Solve a geometry problem involving tangents and secants. Given angles and a tangent line to a circle, find the measure of angle YZ using the Tangent-Secant Theorem and sum of angles in a triangle. Suitable for high school students.

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