Let's analyze the given figure to solve for the values of $x$, $y$, and $z$.

Analysis:

1. Parallelogram Properties:

   • In a parallelogram, opposite angles are equal.
   • Adjacent angles are supplementary (sum up to 180°).

2. Identifying Angles:

   • The given parallelogram is $ABCD$.
   • $\angle EBC$ is given as $40^\circ$.
   • $x$ is the angle at $\angle AEB$.
   • $y$ is the angle at $\angle BEC$.
   • $z$ is the angle at $\angle FDC$.

Step-by-Step Solution:

1. Finding $y$:

   • Since $y$ and the given $40^\circ$ are on a straight line, they are supplementary. $y = 180^\circ - 40^\circ = 140^\circ$

2. Finding $x$:

   • In parallelogram $ABCD$, angle $x$ and angle $y$ are corresponding angles because $AE \parallel DC$ and $EB$ is a transversal. $x = 40^\circ$

3. Finding $z$:

   • $z$ is vertically opposite to angle $x$. Since vertically opposite angles are equal: $z = x = 40^\circ$

Final Answer:

• $x = 40^\circ$
• $y = 140^\circ$
• $z = 40^\circ$

Would you like any further details on this solution, or do you have other questions?

Related Questions:

1. How do the properties of parallelograms help in finding unknown angles?
2. What are vertically opposite angles, and why are they equal?
3. How do supplementary angles work in a geometric figure?
4. Can you explain why adjacent angles in a parallelogram add up to 180 degrees?
5. How would the values change if the given angle was different?

Tip: Always remember that opposite angles in a parallelogram are equal, and adjacent angles are supplementary. This can simplify many angle-finding problems!