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Let's address each question one by one.

### 9) **Prove that the sum of the angles of a triangle is 180°**

To prove this, we can use a simple geometric method. Consider a triangle ABC.

- Draw a line parallel to one side of the triangle (say BC) through the opposite vertex (say A). 
- Let the angles at the base of the triangle be ∠A, ∠B, and ∠C.
- Since the line through A is parallel to BC, by the alternate interior angle theorem, the angle formed at vertex A is equal to the sum of the interior angles at vertices B and C.

Hence, the three interior angles of the triangle sum up to a straight line (180°).

Thus, **∠A + ∠B + ∠C = 180°**, which proves that the sum of the angles in a triangle is always 180 degrees.

### 10) **Find the values of x, y, and z in the adjacent figure**

In the diagram, it is given that **AB ∥ CD** (AB is parallel to CD), and we are asked to find the angles **x°, y°, and z°**.

- **x° (Angle EAB)**: Since AB is parallel to CD and AE is a transversal, the angle **x°** at E is equal to the angle marked as **70°** (corresponding angles on parallel lines are equal).
  - Therefore, **x = 70°**.

- **y° (Angle PAE)**: Angle **PAE** is a right angle (90°), as shown in the figure.
  - Thus, **y = 90°**.

- **z° (Angle DCE)**: Since AB ∥ CD and CE is a transversal, the exterior angle **z°** at C is equal to the interior angle **x°**, again due to the corresponding angles on parallel lines.
  - Thus, **z = 70°**.

### Final Answers:
- **x = 70°**
- **y = 90°**
- **z = 70°**

Would you like further details or have any questions on these problems?

### 5 Related Questions:
1. Can you explain the alternate interior angle theorem in more detail?
2. Why do corresponding angles on parallel lines have equal measures?
3. How can we apply the triangle angle sum property to non-Euclidean geometry?
4. What other properties do triangles have regarding their interior and exterior angles?
5. How can transversal angles be applied in more complex geometric shapes?

### Tip:
When working with parallel lines and transversals, remember that both corresponding angles and alternate interior angles are equal, which is useful for solving various geometry problems.

9) Prove that the sum of the angles of a triangle is 180 degrees. 10) In the adjacent figure, AB is parallel to CD. Find the values of x, y, and z.

Learn how to prove the sum of angles in a triangle is 180 degrees and solve for angles x, y, and z in a geometry problem involving parallel lines and transversals. This explanation uses the Triangle Angle Sum Theorem and Corresponding Angles Postulate.

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