https://file.aiquickdraw.com/mathbot/file/1726008381813grpur9nw.jpg

To solve for \( x \), \( y \), and \( z \) in this diagram, we can use properties of angles formed by intersecting lines.

1. **Vertical angles**: These are angles that are opposite each other when two lines intersect, and they are always congruent (equal in measure).
   - \( x^\circ \) is a vertical angle to \( 43^\circ \), so \( x = 43^\circ \).

2. **Linear pair (supplementary angles)**: Angles that form a straight line add up to \( 180^\circ \).
   - \( y^\circ \) and \( 43^\circ \) form a linear pair, so:
     \[
     y + 43 = 180
     \]
     Solving for \( y \):
     \[
     y = 180 - 43 = 137^\circ
     \]

3. Similarly, \( z^\circ \) and \( x^\circ \) form a linear pair, so:
   \[
   z + x = 180
   \]
   Since \( x = 43^\circ \), we can substitute that in:
   \[
   z + 43 = 180
   \]
   Solving for \( z \):
   \[
   z = 180 - 43 = 137^\circ
   \]

### Final values:
- \( x = 43^\circ \)
- \( y = 137^\circ \)
- \( z = 137^\circ \)

Would you like more details on any of these steps?

Here are 5 related questions to expand on this topic:
1. How are vertical angles defined in terms of intersecting lines?
2. What is the difference between supplementary and complementary angles?
3. Why are the angles \( y \) and \( z \) equal in this case?
4. How would the angle relationships change if the lines were not straight but instead formed a triangle?
5. Can this method be applied to angles formed by parallel lines?

**Tip:** Vertical angles are always equal, no matter the angle size or position of the intersecting lines.

Solve for x, y, and z given the angles in the diagram where one angle is 43 degrees.

Learn how to solve for x, y, and z in a geometry problem involving intersecting lines with one known angle of 43 degrees. This problem covers key geometry concepts like vertical angles and supplementary angles. Suitable for students in Grades 7-9.

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