The task asks to rotate the pentagon $ABCDE$ by 180° clockwise about point $C$, and label the corresponding points $A', B', D', E'$.

Steps to Rotate the Pentagon 180°:

1. Locate point C: Since the rotation is centered around point $C$, this point will remain fixed during the rotation.

2. Determine new positions of other points:

   • For a 180° rotation, each point will end up on the opposite side of $C$, at the same distance from $C$ as before but in the opposite direction.

3. Mapping each vertex:

   • Point A: Count the number of units $A$ is from $C$. It is 3 units right and 2 units up. After the 180° rotation, $A'$ will be 3 units left and 2 units down from $C$.
   • Point B: $B$ is 2 units right and 2 units up from $C$. After the rotation, $B'$ will be 2 units left and 2 units down from $C$.
   • Point D: $D$ is 2 units left and 1 unit down from $C$. After the rotation, $D'$ will be 2 units right and 1 unit up from $C$.
   • Point E: $E$ is 1 unit left and 2 units down from $C$. After the rotation, $E'$ will be 1 unit right and 2 units up from $C$.

Final Positions:

• $A' = (-3, -2)$
• $B' = (-2, -2)$
• $D' = (2, 1)$
• $E' = (1, 2)$

Would you like me to provide more details on any step?

Here are 5 follow-up questions to expand on this:

1. What is the general rule for rotating a point 180° around another point?
2. How can we apply matrix transformations for such rotations?
3. How do clockwise and counterclockwise rotations differ geometrically?
4. Can you explain the symmetry observed in a 180° rotation?
5. What other geometric properties remain unchanged during a rotation?

Tip: When rotating, always ensure to measure distances carefully from the center of rotation for accurate placement.