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To solve this geometry problem, let's break down the given information and the diagram:

### Given Information:
- \( m \angle X = 30^\circ \)
- \( m \angle YT = 50^\circ \)
- \( m \angle WRZ = 90^\circ \)
- \( \overline{XY} \) is tangent to the circle at \( T \)

### Problem Statement:
We need to find the following angles:
- \( m \angle YZ \)
- \( m \angle WZ \)
- \( m \angle TW \)

### Analyzing the Diagram:
1. **Tangent-Secant Theorem**: Since \( \overline{XY} \) is tangent at point \( T \), the angle between the tangent and a chord through the point of tangency is equal to the angle subtended by the chord on the opposite side of the circle. This means:
   \[
   m \angle YXT = m \angle WRZ
   \]
   Since \( m \angle WRZ = 90^\circ \), \( m \angle YXT \) is also \( 90^\circ \).

2. **Finding \( m \angle YZ \)**:
   In triangle \( YTZ \), the sum of the interior angles is \( 180^\circ \). We know:
   \[
   m \angle YT = 50^\circ \quad \text{(given)}
   \]
   Using the tangent-secant property, \( m \angle YTZ = 90^\circ \) (since the tangent line and chord create this angle).
   \[
   m \angle YZ = 180^\circ - (m \angle YT + m \angle YTZ) = 180^\circ - (50^\circ + 90^\circ) = 40^\circ
   \]

3. **Finding \( m \angle WZ \)**:
   Since \( m \angle WRZ = 90^\circ \) and it's a right triangle, and the sum of angles in a triangle is \( 180^\circ \), we can subtract the given angles to find \( m \angle WZ \). In triangle \( WRZ \), we have:
   \[
   m \angle WRZ = 90^\circ
   \]
   Therefore, \( m \angle WZR = 90^\circ - 30^\circ = 60^\circ \).

4. **Finding \( m \angle TW \)**:
   Since \( m \angle WRZ = 90^\circ \) and we already know the angles of the right triangle, the third angle will complete the sum to \( 180^\circ \). Thus,
   \[
   m \angle TW = 180^\circ - 90^\circ - 60^\circ = 30^\circ
   \]

### Final Answer:
- \( m \angle YZ = 40^\circ \)
- \( m \angle WZ = 60^\circ \)
- \( m \angle TW = 30^\circ \)

Would you like further clarification or a detailed explanation of any part? Here are some related questions:

1. How is the tangent-secant angle theorem applied in this problem?
2. Why is \( \angle WRZ \) equal to \( 90^\circ \)?
3. How do you calculate the angles in a right triangle?
4. What is the sum of the angles in a triangle, and how is it used here?
5. How does the angle at the tangent point influence the other angles in the circle?

**Tip**: Remember, in any triangle, the sum of the angles is always \( 180^\circ \), and this is key when dealing with circles and tangents.

Resuelve esto: m∠X = 30, m∠YT = 50, m∠WRZ = 90, XY es tangente al círculo. Encuentra: m∠YZ, m∠WZ y m∠TW.

Solve a geometry problem involving a tangent to a circle and a secant. Given angles m∠X = 30°, m∠YT = 50°, and m∠WRZ = 90°, calculate the angles m∠YZ, m∠WZ, and m∠TW using the tangent-secant theorem and properties of triangles. Suitable for Grades 9-11.

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