To solve for the angles $x$, $y$, $z$, and $a$ in the given parallelograms, let's analyze both parallelograms separately:

Parallelogram $a$:

1. Properties of a Parallelogram:
   • Opposite angles are equal.
   • Adjacent angles are supplementary (they add up to $180^\circ$).

Given:

• $\angle A = 60^\circ$
• $\angle B = 120^\circ$

From the properties of parallelograms:

• $\angle D = \angle B = 120^\circ$ (opposite angles are equal)
• $\angle C = \angle A = 60^\circ$ (opposite angles are equal)

Now, the angles $x$, $y$, $z$, and $a$ correspond to:

• $x = \angle C = 60^\circ$
• $y = \angle D = 120^\circ$
• $z = \angle B = 120^\circ$
• $a = \angle A = 60^\circ$

Parallelogram $b$:

Given:

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

In this parallelogram, let's first verify the missing angle measures:

• The angles in a quadrilateral must sum to $360^\circ$.

The sum of the given angles: $\angle S + \angle R + \angle P + \angle Q = 360^\circ$

Thus: $45^\circ + 25^\circ + \angle P + \angle Q = 360^\circ$ $70^\circ + \angle P + \angle Q = 360^\circ$ $\angle P + \angle Q = 290^\circ$

But we know that in a parallelogram, opposite angles are equal, and adjacent angles are supplementary:

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

Thus:

• $x = \angle Q = 45^\circ$
• $y = \angle R = 25^\circ$
• $z = \angle S = 45^\circ$
• $a = \angle P = 25^\circ$

Final Answers:

1. For Parallelogram $a$:
   • $x = 60^\circ$
   • $y = 120^\circ$
   • $z = 120^\circ$
   • $a = 60^\circ$
2. For Parallelogram $b$:
   • $x = 45^\circ$
   • $y = 25^\circ$
   • $z = 45^\circ$
   • $a = 25^\circ$

Would you like further details on any step?

Here are some follow-up questions for you:

1. What other properties can be used to determine angles in parallelograms?
2. How can you prove that opposite angles in a parallelogram are equal?
3. What other quadrilaterals have properties similar to parallelograms?
4. How can the diagonals of a parallelogram help in finding the angles?
5. What role does symmetry play in determining the angles of special parallelograms like rhombuses or rectangles?

Tip: Always double-check that the sum of the angles in a quadrilateral equals $360^\circ$ to ensure accuracy in your solutions!