We are given that triangle $\triangle CDE$ is isosceles, meaning two of its sides are congruent: $EC \cong DE$. Additionally, we know that $m\angle D = 72^\circ$.

In an isosceles triangle, the base angles (the angles opposite the congruent sides) are equal. Therefore:

• $\angle C$ is congruent to $\angle E$, i.e., $m\angle C = m\angle E$.

Now, the sum of the interior angles of any triangle is $180^\circ$. So, we can write:

$m\angle C + m\angle E + m\angle D = 180^\circ$

Since $m\angle C = m\angle E$, let’s denote $m\angle C = x$. The equation becomes:

$x + x + 72^\circ = 180^\circ$

Simplifying:

$2x + 72^\circ = 180^\circ$

Subtract $72^\circ$ from both sides:

$2x = 108^\circ$

Now, divide by 2:

$x = 54^\circ$

Thus, $m\angle C = 54^\circ$.

Final Answer:

$m\angle C = 54^\circ$

Would you like more details on the solution? Here are some related questions:

1. How do you determine the angles in an isosceles triangle?
2. Why is the sum of the angles in a triangle always $180^\circ$?
3. Can you find the measure of $m\angle E$ in this triangle?
4. How would the result change if $m\angle D$ was different?
5. What happens if all sides of the triangle are congruent?

Tip: Always remember that in any triangle, the sum of all interior angles equals 180 degrees!