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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.

Explore how to find the measures of angles x, y, z, and a in parallelogram PQRS with a detailed step-by-step solution. Learn about the properties of parallelograms, including opposite angles and adjacent angles. Suitable for students in Grades 7-9 studying geometry and angles.

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