Let’s carefully solve for $m\angle C$ using the information given and the properties of the figure in the diagram:

Step-by-step Solution:

1. Identifying Key Elements:

   • $DE \cong DF$: $DE$ and $DF$ are congruent, indicating that $\triangle DEF$ is an isosceles triangle.
   • $CD \parallel BE$ and $BC \parallel FD$: This implies that $BCDF$ is a parallelogram.
   • $m\angle FDE = 56^\circ$: The measure of angle $FDE$ is $56^\circ$.

2. Analyzing $\triangle DEF$: Since $\triangle DEF$ is isosceles with $DE = DF$, the base angles $\angle DEF$ and $\angle DFE$ are equal.

   • Given that $m\angle FDE = 56^\circ$, we can find the other two angles of the triangle: $m\angle DEF = m\angle DFE.$

3. Sum of Angles in a Triangle: The sum of the angles in a triangle is $180^\circ$. So, for $\triangle DEF$: $m\angle FDE + m\angle DEF + m\angle DFE = 180^\circ.$ Plugging in the values: $56^\circ + 2 \times m\angle DEF = 180^\circ,$ $2 \times m\angle DEF = 180^\circ - 56^\circ,$ $2 \times m\angle DEF = 124^\circ,$ $m\angle DEF = 62^\circ.$

4. Using Parallel Lines in Parallelogram $BCDF$: Since $BC \parallel FD$ and $CD \parallel BE$, we know that:

   • $m\angle C$ (or $m\angle DFE$) is the corresponding angle with $m\angle DEF$ due to the parallelism between $BC$ and $FD$.

   Therefore: $m\angle C = m\angle DFE = 62^\circ.$

Final Answer:

$\boxed{m\angle C = 62^\circ}$

Do you need further clarification or have any questions?

Related Questions:

1. What is the significance of parallel lines in determining angle measures?
2. How do the properties of an isosceles triangle influence its angle calculations?
3. How can identifying a parallelogram simplify angle problems?
4. What other properties of parallelograms are useful in geometry problems?
5. How do you solve for unknown angles in complex geometric shapes?

Tip:

Remember that in parallelograms, opposite angles are equal, and consecutive angles are supplementary, which can help when solving for unknowns!