https://file.aiquickdraw.com/mathbot/file/172652068375445s4kkrk.jpg

In this image, you have a triangle with the following information:

- \( \angle X\hat{P}Z = 100^\circ \)
- \( \angle Z\hat{Y}P = 20^\circ \)

We are required to find the values of \( x \) and \( y \), which refer to the angles \( \angle X\hat{Z}Y \) and \( \angle Y\hat{Z}P \), respectively.

### Step-by-step solution:

1. **Understanding the triangle:**
   - \( YZ = XZ \) (because of the equal markings on the sides, indicating that it is an isosceles triangle).
   - Hence, \( \angle X\hat{Z}Y = \angle Y\hat{Z}P = x \), which is one of the key points we will use.

2. **Sum of angles around point \( Z \):**
   Since the angle \( \angle X\hat{P}Z = 100^\circ \), and we have two angles \( x \) and \( y \), the sum of angles in any triangle is \( 180^\circ \).
   So, for \( \triangle XYZ \), we have:
   \[
   x + x + 100^\circ = 180^\circ
   \]
   This simplifies to:
   \[
   2x + 100^\circ = 180^\circ
   \]
   \[
   2x = 80^\circ
   \]
   \[
   x = 40^\circ
   \]

3. **Now solve for \( y \):**
   Since \( \angle Z\hat{Y}P = 20^\circ \) and the angles at point \( Z \) must sum up to \( 180^\circ \), we use the following:
   \[
   x + y + 20^\circ = 180^\circ
   \]
   Substituting \( x = 40^\circ \):
   \[
   40^\circ + y + 20^\circ = 180^\circ
   \]
   \[
   y = 180^\circ - 60^\circ = 120^\circ
   \]

### Final Answer:
- \( x = 40^\circ \)
- \( y = 120^\circ \)

Would you like a detailed explanation of any step or have any other questions? Here are five related questions you might find useful:

1. How do we identify when a triangle is isosceles?
2. What are the key properties of isosceles triangles?
3. Can the exterior angle of a triangle help in solving for unknown angles?
4. How do we apply the angle sum property in non-right triangles?
5. What are some methods to solve for angles in scalene triangles?

**Tip:** Always check for symmetry or congruent sides in a triangle to simplify calculations.

X̂PZ = 100°, ẐYP = 20°, determine the size of x and y

Learn how to find the angles x and y in triangle X̂PZ where the given angles are X̂PZ = 100° and ẐYP = 20°. This geometry problem covers key concepts like isosceles triangles and the angle sum property, suitable for Grades 9-10.

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