To determine the measures of angles $P$ and $S$, let’s analyze the given parallelogram.

Key Properties of a Parallelogram:

1. Opposite angles are equal.
2. Adjacent angles are supplementary (they add up to $180^\circ$).

From the diagram:

• Angle $Q = (x + 15)^\circ$
• Angle $R = (6x - 10)^\circ$

Step 1: Find $x$

Since $Q$ and $R$ are adjacent angles of the parallelogram, they are supplementary: $\text{Angle } Q + \text{Angle } R = 180^\circ$ $(x + 15) + (6x - 10) = 180$

Simplify: $x + 15 + 6x - 10 = 180$ $7x + 5 = 180$ $7x = 175$ $x = 25$

Step 2: Find the measure of angles $Q$ and $R$

Substitute $x = 25$ into the expressions for $Q$ and $R$:

• $Q = x + 15 = 25 + 15 = 40^\circ$
• $R = 6x - 10 = 6(25) - 10 = 150 - 10 = 140^\circ$

Step 3: Determine angles $P$ and $S$

• $P$ is opposite to $R$, so $\angle P = 140^\circ$ (opposite angles are equal).
• $S$ is opposite to $Q$, so $\angle S = 40^\circ$.

Final Answer:

$\angle P = 140^\circ \quad \text{and} \quad \angle S = 40^\circ$

Correct option: ∠P = 140°; ∠S = 40°

Do you need further clarification on any step? Let me know!

Related Questions:

1. How do you find the missing angles in a parallelogram?
2. What are the properties of a parallelogram related to angles and sides?
3. How do you solve for $x$ in geometric problems with supplementary angles?
4. Can you verify the solution by using a different approach?
5. How are diagonals in a parallelogram related to the angles?

Tip:

In any parallelogram, always check adjacent and opposite angles carefully—they follow specific relationships that simplify the solution process.