To solve the problem, we use the given information:

1. AB is parallel to EF ($AB \parallel EF$): This implies that certain angle relationships arise due to parallel lines, such as corresponding angles and supplementary angles.

2. The angles provided:

   • $\angle CAB = 228^\circ$
   • $\angle EDF = 293^\circ$
   • $\angle CEF = 32^\circ$
   • We are solving for $b^\circ$ (angle $\angle DCE$).

Step-by-Step Solution:

Step 1: Analyze supplementary angles on a straight line.

• At point $A$, the angle $\angle CAB = 228^\circ$ exceeds $180^\circ$, so it spans both the exterior and interior regions. This means the reflex part is: \text{Interior angle at A} = 228^\circ - 180^\circ = 48^\circ.

Step 2: Use parallel line properties and the transversal.

• The interior angle at $A$ ($48^\circ$) corresponds to the angle at $E$ on the opposite side of the transversal, due to alternate interior angles. So: $\angle CEF = 48^\circ.$

Step 3: Solve for $b^\circ$ using triangle angle sum.

• In $\triangle CDE$, the sum of angles is $180^\circ$. The known angles are:
  • $\angle CEF = 32^\circ$ (given),
  • $\angle CED = 293^\circ - 180^\circ = 113^\circ$ (interior part of the reflex angle at $D$).
• Using the triangle angle sum property: $b^\circ + 32^\circ + 113^\circ = 180^\circ.$

Step 4: Simplify for $b$.

$b^\circ = 180^\circ - 113^\circ - 32^\circ = 35^\circ.$

Final Answer:

$b = 35^\circ$

Let me know if you need further clarifications or assistance! Here are some follow-up questions to deepen understanding:

1. What are the alternate interior angle and corresponding angle properties in parallel lines?
2. Why is the triangle angle sum always $180^\circ$?
3. Can you identify another angle in the diagram if one more is missing?
4. How do reflex angles work in geometry?
5. Why does $AB \parallel EF$ lead to specific angle relationships?

Tip: When working with parallel lines, always look for transversals and use properties like corresponding and alternate interior angles!