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The given problem involves two parallel lines and the goal is to prove that two angles are complementary. Let's break down the key steps of the proof so far and continue solving it:

### Problem Details:
- **Given:**
  1. \( BC \parallel DE \)
  2. \( \angle ABC \) and \( \angle AED \) are complementary (i.e., their sum is \( 90^\circ \)).
  
- **To Prove:**
  \( \angle ABC \) and \( \angle ACB \) are complementary.

### Proof Breakdown:
- **Step 1:**
  - **Statement:** \( BC \parallel DE \)
  - **Reason:** Given.
  
- **Step 2:**
  - **Statement:** \( \angle ABC \) and \( \angle AED \) are complementary.
  - **Reason:** Given.
  
- **Step 3:**
  - **Statement:** \( \angle ADE \cong \angle ABC \).
  - **Reason:** Parallel lines cut by a transversal form corresponding angles, so these two angles are congruent.

### Continuing the Proof:
- **Step 4:** 
  Since \( \angle ABC \) and \( \angle ADE \) are congruent, and \( \angle ABC \) and \( \angle AED \) are complementary, this implies that \( \angle ADE \) and \( \angle ACB \) are also complementary by substitution. 

- **Reason:** If two angles are congruent and complementary to a third angle, they are complementary to each other.

Thus, \( \angle ABC \) and \( \angle ACB \) are complementary, completing the proof.

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Would you like more details on any step, or is there any part you'd like me to clarify?

### Related Questions:
1. What does it mean for two angles to be complementary?
2. How do corresponding angles behave when parallel lines are cut by a transversal?
3. How can you use the property of congruent angles in geometric proofs?
4. What is the significance of transversals in geometric proofs?
5. Why can't quadrilateral properties be used in this proof?

**Tip:** When working with parallel lines and transversals, remember that corresponding, alternate interior, and alternate exterior angles have special relationships.

Given BC || DE and ∠ABC and ∠AED are complementary, prove ∠ABC and ∠ACB are complementary.

Learn how to prove that ∠ABC and ∠ACB are complementary by using the properties of parallel lines cut by a transversal. This proof involves the corresponding angles theorem and complementary angles. Suitable for high school students in grades 9-10.

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