To find the values of $x$, $y$, and $z$, let's analyze the geometry and angles in the diagram:

1. Angle $x$:

   • The sum of angles on a straight line is $180^\circ$.
   • The angles on the straight line adjacent to $x$ are $45^\circ$ and $66^\circ$.
   • Therefore, $x + 45^\circ + 66^\circ = 180^\circ$.
   • Simplifying, $x = 180^\circ - (45^\circ + 66^\circ)$.
   • $x = 180^\circ - 111^\circ$.
   • $x = 69^\circ$.

2. Angle $y$:

   • The sum of angles in a triangle is $180^\circ$.
   • The angles in the triangle adjacent to $y$ are $30^\circ$ and $43^\circ$.
   • Therefore, $y + 30^\circ + 43^\circ = 180^\circ$.
   • Simplifying, $y = 180^\circ - (30^\circ + 43^\circ)$.
   • $y = 180^\circ - 73^\circ$.
   • $y = 107^\circ$.

3. Angle $z$:

   • The sum of angles on a straight line is $180^\circ$.
   • The angles on the straight line adjacent to $z$ are $119^\circ$ and $y$ (which we just found to be $107^\circ$).
   • Therefore, $z + 119^\circ + 107^\circ = 180^\circ$.
   • Simplifying, $z = 180^\circ - (119^\circ + 107^\circ)$.
   • $z = 180^\circ - 226^\circ$.
   • $z = -46^\circ$.

This indicates a problem: the values we have obtained don't seem to align with the typical rules of geometry. Let's revisit and verify the values of angles more carefully:

1. Angle $x$ (Second Attempt):

   • Correct expression: $x + 45^\circ + 66^\circ = 180^\circ$.
   • Simplifying: $x = 180^\circ - 111^\circ = 69^\circ$.

2. Angle $y$ (Second Attempt):

   • Correct expression: $y + 30^\circ + 43^\circ = 180^\circ$.
   • Simplifying: $y = 180^\circ - 73^\circ = 107^\circ$.

3. Angle $z$ (Correction):

   • If angle $y$ is part of a straight line with $z$ and $119^\circ$, the equation is:
   • $y + z = 180^\circ$.
   • Substituting $y = 107^\circ$: $107^\circ + z = 180^\circ$.
   • $z = 180^\circ - 107^\circ = 73^\circ$.

Correct values:

• $x = 69^\circ$
• $y = 107^\circ$
• $z = 73^\circ$

Would you like a more detailed explanation or have any further questions?

Related Questions:

1. How do you find the sum of angles in a triangle?
2. What is the property of angles on a straight line?
3. How do you determine unknown angles using supplementary angles?
4. Can two triangles have the same angles but different side lengths?
5. What is the difference between complementary and supplementary angles?

Tip: When solving geometric problems, always double-check which angles form linear pairs and ensure the sum of angles in triangles and lines matches the expected values.