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To determine the measures of angles \( x \), \( y \), and \( z \) in the diagram, let's analyze the information given:

1. **Triangle with angles 58° and 83°**:
   - The sum of angles in a triangle is 180°.
   - Therefore, the third angle in this triangle (which is \( x \)) can be calculated as:
     \[
     x = 180^\circ - 58^\circ - 83^\circ = 39^\circ
     \]

2. **Angle relationships around point where \( y \) and \( z \) are located**:
   - The angles \( x \), \( y \), and 103° form a straight line.
   - Therefore, the sum of these three angles is 180°:
     \[
     x + y + 103^\circ = 180^\circ
     \]
   - Substitute \( x = 39^\circ \) from step 1:
     \[
     39^\circ + y + 103^\circ = 180^\circ
     \]
     \[
     y = 180^\circ - 142^\circ = 38^\circ
     \]

3. **Triangle with angles 63° and \( y \)**:
   - Since \( y = 38^\circ \), we can find \( z \) in this triangle using the sum of angles in a triangle:
     \[
     z = 180^\circ - 63^\circ - 38^\circ = 79^\circ
     \]

### Summary of angle measures:
- \( x = 39^\circ \)
- \( y = 38^\circ \)
- \( z = 79^\circ \)

Would you like further details on each step, or do you have any specific questions?

Here are some related questions to deepen your understanding:
1. How do you determine the third angle in any triangle if two angles are known?
2. What is the significance of a linear pair of angles in geometry?
3. How does the exterior angle of a triangle relate to its interior opposite angles?
4. Why is the sum of angles in a triangle always 180°?
5. How would you solve for an unknown angle if given multiple triangles connected by shared angles?

**Tip:** Remember, in any triangle, the sum of the interior angles is always 180°. This rule is foundational and useful in solving many geometric problems.

Determine the measures of angles x, y, and z in the given geometric diagram.

Solve for angles x, y, and z in a complex geometric diagram using the angle sum theorem and linear pair theorem. Detailed explanation of key geometry concepts suitable for students in Grades 7-9.

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