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.