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已知：AB=CD，角B为40°，角ADC为70°，求角C

In the image you provided, there's a triangle \( ABC \) with \( AB = DC \) and the following angles: \( \angle B = 40^\circ \) and \( \angle ADC = 70^\circ \). We are asked to find \( \angle C \).

Here's how to solve it:

1. **Recognize that \( \triangle ABD \) and \( \triangle ACD \) are isosceles:** Since \( AB = DC \), and \( D \) is on \( AC \), both \( \triangle ABD \) and \( \triangle ACD \) are isosceles. Thus, \( \angle ADB = \angle ABD \) and \( \angle ADC = \angle ACD \).

2. **Determine \( \angle ADB \) and \( \angle ABD \):**
   - Since \( \angle ADC = 70^\circ \) and \( \triangle ACD \) is isosceles, \( \angle ACD = \angle ADC = 70^\circ \).
   - \( \angle ABD = \angle B = 40^\circ \).
   - \( \angle ADB \) can be determined as follows:
     \[
     \angle ADB = 180^\circ - (\angle ABD + \angle ACD) = 180^\circ - (40^\circ + 70^\circ) = 70^\circ.
     \]

3. **Calculate \( \angle BAC \):**
   \[
   \angle BAC = \angle ADB + \angle ABD = 70^\circ + 40^\circ = 110^\circ.
   \]

4. **Find \( \angle C \):**
   \[
   \angle C = 180^\circ - \angle BAC = 180^\circ - 110^\circ = 70^\circ.
   \]

Thus, \( \angle C = 70^\circ \).

Would you like more details or have any other questions on this? Here are some follow-up questions to consider:

1. How does the property of isosceles triangles help in solving problems like these?
2. What would change if \( AB \neq DC \)?
3. Could this problem be solved using trigonometry instead?
4. What is the significance of point \( D \) in this triangle configuration?
5. How would the angles change if \( \angle B \) was greater than \( 40^\circ \)?
6. What are some other properties of isosceles triangles?
7. Can you explain the difference between isosceles and equilateral triangles?
8. What happens to \( \angle C \) if \( \angle ADC \) increases or decreases?

And a tip: Remember that in any triangle, the sum of the interior angles is always \( 180^\circ \). This fundamental property can often help simplify and solve angle-related problems in geometry.

Learn how to find angle C in triangle ABC where AB = CD, angle B = 40°, and angle ADC = 70°. This problem involves the properties of isosceles triangles and the angle sum property of triangles. Suitable for high school students studying geometry.

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