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In this problem, we are given a geometric diagram with certain angle relationships, and we need to find \( m\angle C \).

### Information Provided:
- \( \angle B \cong \angle BAD \cong \angle ADC \)
- \( m\angle ADE = 80^\circ \)
- We need to find \( m\angle C \).

Let's break down the problem:

### Step 1: Identify the Given Angles
- \( m\angle ADE = 80^\circ \)
- \( \angle B \cong \angle BAD \cong \angle ADC \)

From this, we can conclude that:
- \( \angle B = \angle BAD = \angle ADC \)

### Step 2: Understand the Structure of the Diagram
Since the three angles \( \angle B \), \( \angle BAD \), and \( \angle ADC \) are congruent, they must each have the same measure.

Let the measure of these angles be denoted by \( x \). So:
\[ m\angle B = m\angle BAD = m\angle ADC = x \]

### Step 3: Use Triangle Angle Sum
Notice that \( \triangle ADE \) is a triangle, and we already know:
- \( m\angle ADE = 80^\circ \)
- \( m\angle BAD = x \)
- The third angle, \( \angle EAD \), must satisfy the triangle sum property, which states that the sum of the interior angles of a triangle is \( 180^\circ \).

Thus, we can write the equation:
\[
x + 80^\circ + \angle EAD = 180^\circ
\]
This simplifies to:
\[
\angle EAD = 180^\circ - 80^\circ - x = 100^\circ - x
\]

### Step 4: Apply to Triangle ABC
Now let's focus on triangle \( ABC \), where we need to find \( m\angle C \).

Since \( \angle B = x \) and \( \angle ADC = x \), we can use the fact that the sum of the angles in a triangle is \( 180^\circ \). For \( \triangle ABC \), the angles are:
\[
m\angle B + m\angle A + m\angle C = 180^\circ
\]
Substituting \( m\angle B = x \) and \( m\angle A = x \), we get:
\[
x + x + m\angle C = 180^\circ
\]
This simplifies to:
\[
2x + m\angle C = 180^\circ
\]
Solving for \( m\angle C \), we get:
\[
m\angle C = 180^\circ - 2x
\]

### Step 5: Conclusion
To find the exact value of \( m\angle C \), we need to determine the value of \( x \), which could be done with additional geometric information or reasoning. Based on the diagram, angle relationships, and congruency statements, you can find \( m\angle C \).

Would you like me to continue with more details or assumptions to finalize this solution? Let me know if you have any questions so far. Here are 5 related questions:

1. How do we determine congruent angles in geometric diagrams?
2. What is the significance of the triangle angle sum property?
3. How can we apply angle bisector theorems in geometry?
4. How do parallel lines influence angle measures in such diagrams?
5. What methods can be used to calculate unknown angles when only one is given?

**Tip**: Always double-check angle congruencies, as they are key in solving geometric problems!

In the diagram, \(\angle B \cong \angle BAD \cong \angle ADC\) and \(m\angle ADE = 80^\circ\). Find \(m\angle C\).

Learn how to find the measure of angle C in a geometric diagram where angles B, BAD, and ADC are congruent, and the angle ADE is 80°. Using triangle angle sum properties and congruencies, we solve for the unknown angle.

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