To solve for the angles given in the diagram, let's analyze the relationships between the lines, angles, and the provided $65^\circ$ angle:

a) $\angle DRS$:

The lines $DR$ and $ST$ are parallel, and $RS$ is a transversal line.

By the property of alternate interior angles: $\angle DRS = \angle QRS = 65^\circ$ Answer: $\angle DRS = 65^\circ$

b) $\angle QRC$:

Since $DR \parallel ST$ and $RS$ is a transversal, $\angle QRC$ is supplementary to $\angle QRS$ (linear pair of angles).

$\angle QRC = 180^\circ - 65^\circ = 115^\circ$ Answer: $\angle QRC = 115^\circ$

c) $\angle DRQ$:

Here, $DR \parallel ST$ and $PQ$ is a transversal. $\angle DRQ$ forms a corresponding angle with $\angle PQB$ (which is $65^\circ$).

$\angle DRQ = 65^\circ$ Answer: $\angle DRQ = 65^\circ$

d) $\angle RSE$:

Since $RS$ is a transversal and $DR \parallel ST$, $\angle RSE$ forms a corresponding angle with $\angle QRS$.

$\angle RSE = 65^\circ$ Answer: $\angle RSE = 65^\circ$

e) $\angle FST$:

$FST$ is a straight angle (on the line $ST$), so it measures:

$\angle FST = 180^\circ$ Answer: $\angle FST = 180^\circ$

f) $\angle RSF$:

$\angle RSF$ and $\angle RSE$ form a linear pair of angles. Therefore:

$\angle RSF = 180^\circ - 65^\circ = 115^\circ$ Answer: $\angle RSF = 115^\circ$

Final Answers Summary:

a) $\angle DRS = 65^\circ$
b) $\angle QRC = 115^\circ$
c) $\angle DRQ = 65^\circ$
d) $\angle RSE = 65^\circ$
e) $\angle FST = 180^\circ$
f) $\angle RSF = 115^\circ$

Do you need explanations for any specific step? Here are 5 related questions:

1. What are alternate interior angles, and how are they identified?
2. How do you determine corresponding angles with parallel lines?
3. Why do linear pairs of angles always add up to $180^\circ$?
4. What is the difference between supplementary and complementary angles?
5. How can transversal lines create different angle relationships with parallel lines?

Tip: Always look for parallel lines and transversal relationships when solving for angles in geometry diagrams!