To solve for the measures of $a$ and $b$ in the given geometry problem, we'll use the properties of parallel lines and the angles formed by a transversal. Here’s a step-by-step approach:

Step 1: Identify Angle Relationships

• Since $PQ \parallel RS$, the transversal creates corresponding, alternate, and co-interior angles between these lines.
• Angle $e$ is given as $120^\circ$.
• Angle $f$ is $65^\circ$.

Step 2: Solve for Angle $a$

• $a$ is vertically opposite to the angle $e$, so: $a = 120^\circ$ because vertically opposite angles are equal.

Step 3: Solve for Angle $b$

• $b$ and angle $f$ are co-interior (consecutive interior) angles with respect to parallel lines $PQ \parallel RS$ and transversal $PR$. Therefore, they sum to $180^\circ$: $b + 65^\circ = 180^\circ$
• Solving for $b$: $b = 180^\circ - 65^\circ = 115^\circ$

Final Answer:

• $a = 120^\circ$
• $b = 115^\circ$

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

Here are 5 related questions:

1. How do you determine corresponding angles when parallel lines are cut by a transversal?
2. What are alternate interior angles, and how are they identified?
3. Can two angles be supplementary without being on the same line?
4. What is the difference between vertically opposite angles and linear pairs?
5. How does the parallel postulate help in solving problems like this one?

Tip: When solving problems involving parallel lines and transversals, always look for angle pairs that follow known rules like corresponding, alternate interior, or co-interior angles.