In the diagram below, angle, B, \cong, angle, B, A, D, \cong, angle, A, D, C∠B≅∠BAD≅∠ADC and m, angle, A, D, E, equals, 80, degrees, .m∠ADE=80 
∘
 . Find m, angle, C, .m∠C.


In this problem, you're working with a geometric diagram where multiple angles are congruent, and you're asked to find the measure of \( m\angle C \). Here's a step-by-step approach:

1. **Given information**:
   - \( \angle B \cong \angle BAD \cong \angle ADC \), which means these three angles are congruent (equal in measure).
   - \( m\angle ADE = 80^\circ \), meaning the measure of angle \( ADE \) is 80 degrees.

2. **Triangle \( ADE \)**:
   The triangle \( ADE \) has angle \( ADE = 80^\circ \), and the sum of all interior angles in any triangle is always 180 degrees.

3. **Finding the other angles in triangle \( ADE \)**:
   Let's say the measure of angle \( DAE \) is \( x \), then the sum of the interior angles of triangle \( ADE \) is:
   \[
   m\angle ADE + m\angle DAE + m\angle DEA = 180^\circ
   \]
   \[
   80^\circ + x + m\angle DEA = 180^\circ
   \]
   Thus:
   \[
   x + m\angle DEA = 100^\circ
   \]

4. **Congruent angles**:
   Since \( \angle B \cong \angle BAD \cong \angle ADC \), the measures of these angles are equal. If you know or assume any relationships between the diagram's structure or additional angle information, you can proceed to calculate \( m\angle C \).

Without the exact diagram, these steps outline how to approach the solution. Could you confirm the diagram's structure or provide additional details? I’ll assist further based on that.

Let me know if you'd like more details on any part or have any clarifications!

### Related Questions:
1. How are angles \( \angle ADE \) and \( \angle DAE \) related in the triangle?
2. What is the total measure of interior angles in any triangle?
3. How can we use the congruent angles in this problem to find \( m\angle C \)?
4. How does the property of congruent angles help in triangle problems?
5. What would happen if the sum of angles in a triangle was not 180 degrees?

**Tip:** Always check for congruent triangles or isosceles triangles in geometric problems—they often simplify the calculations!

In the diagram below, angle B ≅ angle BAD ≅ angle ADC and m∠ADE = 80°. Find m∠C.

Explore a step-by-step solution to find the measure of angle C (m∠C) in a geometric diagram involving congruent angles B ≅ BAD ≅ ADC and m∠ADE = 80°. Learn about key geometric concepts, including congruent angles and the sum of interior angles in a triangle, suitable for Grades 8-10 students.

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