Let's solve part b of the problem, where we are asked to find the measures of angles $x$, $y$, $z$, and $a$ in the parallelogram $PQRS$.

Step 1: Understanding the properties of a parallelogram

In a parallelogram:

1. Opposite angles are equal.
2. Adjacent angles are supplementary (i.e., they sum up to $180^\circ$).

Step 2: Analyzing the given angles

• $\angle S = 45^\circ$
• $\angle R = 25^\circ$

Step 3: Finding the unknown angles

• Since $\angle P + \angle S = 180^\circ$ (because they are adjacent angles): $\angle P = 180^\circ - \angle S = 180^\circ - 45^\circ = 135^\circ$ Therefore, $x = 135^\circ$.

• Similarly, $\angle Q + \angle R = 180^\circ$: $\angle Q = 180^\circ - \angle R = 180^\circ - 25^\circ = 155^\circ$ Therefore, $y = 155^\circ$.

• Since opposite angles in a parallelogram are equal: $z = \angle S = 45^\circ$ $a = \angle R = 25^\circ$

Final Answers:

• $x = 135^\circ$
• $y = 155^\circ$
• $z = 45^\circ$
• $a = 25^\circ$

Would you like details on any step or have any other questions?

Here are some related questions you might find interesting:

1. How do adjacent angles in a parallelogram relate to each other?
2. Why do opposite angles in a parallelogram have equal measures?
3. How would the angle measures change if this were a rectangle instead of a parallelogram?
4. How can you prove that a quadrilateral is a parallelogram based on its angles?
5. Can you calculate the area of a parallelogram if you know all angles and one side length?

Tip: In any parallelogram, understanding the relationship between adjacent and opposite angles helps solve most angle-related problems quickly.