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The problem you've provided involves a triangle \( XYZ \) where point \( A \) is on side \( YZ \) such that \( XA = YA \). Given:

- \( \angle XYA = 50^\circ \)
- \( \angle AXZ = 19^\circ \)

We are asked to find the degree measure of \( \angle XZA \).

Since \( XA = YA \), triangle \( XYA \) is isosceles, which means \( \angle XAY = \angle XYA = 50^\circ \). The sum of the interior angles of a triangle is \( 180^\circ \), so we can calculate \( \angle AXY \) in triangle \( XYA \).

\[
\angle AXY = 180^\circ - \angle XYA - \angle XAY = 180^\circ - 50^\circ - 50^\circ = 80^\circ
\]

Now, consider triangle \( AXZ \). We know that:

- \( \angle AXZ = 19^\circ \)
- \( \angle XAZ = \angle XYA = 50^\circ \)

To find \( \angle XZA \), we use the fact that the angles of triangle \( AXZ \) must also sum to \( 180^\circ \):

\[
\angle XZA = 180^\circ - \angle AXZ - \angle XAZ = 180^\circ - 19^\circ - 50^\circ = 111^\circ
\]

However, none of the provided answer choices match this directly, so it might involve more geometric analysis or the wording of the problem could suggest a different interpretation. Do you want further clarification or exploration into why this discrepancy might arise?

Here are some further questions to explore:

1. How does the isosceles triangle property \( XA = YA \) influence other parts of the solution?
2. Why is the sum of the angles in triangle \( AXZ \) important for calculating \( \angle XZA \)?
3. Could there be any additional geometric theorems, such as angle bisectors, involved in this problem?
4. How does the structure of the problem change if point \( A \) lies on the exterior of the triangle?
5. What role do supplementary angles play in understanding this triangle?

**Tip:** Always verify your steps against the given choices and reconsider geometric relationships for possible errors or alternative configurations.

In triangle XYZ, A is a point on YZ such that XA = YA. If ∠XYA = 50° and ∠AXZ = 19°, what is the degree measure of ∠XZA?

Learn how to solve for the angle XZA in triangle XYZ where XA = YA, and angles XYA and AXZ are given. This problem covers key geometric concepts, including the isosceles triangle theorem and the sum of interior angles in a triangle. Suitable for students in Grades 9-10.

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