In triangle, C, D, E, comma△CDE, start overline, E, C, end overline, \cong, start overline, D, E, end overline 
EC
 ≅ 
DE
  and m, angle, D, equals, 72, degrees, .m∠D=72 
∘
 . Find m, angle, C, .m∠C.

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!

In triangle CDE, EC ≅ DE and m∠D = 72°. Find m∠C.

Learn how to find the measure of angle C in an isosceles triangle CDE, where EC ≅ DE and m∠D = 72°. This step-by-step solution covers key geometric concepts such as the Isosceles Triangle Theorem and the angle sum property. Suitable for Grades 7-9.

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