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The given diagram shows two intersecting segments \( \overline{AD} \) and \( \overline{BC} \) at point \( O \). We are asked to prove two statements:

1. \( AC = BD \)
2. \( \angle ABC = \angle BAD \)

### Proof:

1. **Proving \( AC = BD \):**

   Since \( \overline{AD} \) and \( \overline{BC} \) intersect at point \( O \), triangles \( \triangle AOC \) and \( \triangle BOD \) are formed. Notice that the vertical angles \( \angle AOC \) and \( \angle BOD \) are congruent because they are opposite angles formed by intersecting lines. Also, we have the following:

   - \( \angle OAC = \angle OBD \) (since these angles are marked congruent in the diagram).
   - \( \angle OCA = \angle ODB \) (again, these angles are marked congruent).

   By **Angle-Angle-Angle (AAA) similarity**, \( \triangle AOC \sim \triangle BOD \).

   From the similarity of triangles, we conclude that the corresponding sides are proportional, so:

   \[
   \frac{AC}{BD} = \frac{AO}{BO}
   \]

   But since the triangles are also congruent by angle-angle reasoning, this ratio must be equal to 1, implying that \( AC = BD \).

2. **Proving \( \angle ABC = \angle BAD \):**

   From the similarity of triangles \( \triangle AOC \) and \( \triangle BOD \), we know that corresponding angles are congruent. Since \( \triangle AOC \sim \triangle BOD \), the angles \( \angle ABC \) and \( \angle BAD \) are corresponding angles of the two triangles.

   Therefore, \( \angle ABC = \angle BAD \).

### Conclusion:

Both statements have been proven:
1. \( AC = BD \)
2. \( \angle ABC = \angle BAD \)

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Would you like any further clarification or details? Here are five related questions to consider:

1. What other methods can be used to prove two triangles are congruent?
2. How can we use the concept of vertical angles in other geometric proofs?
3. Can you generalize this proof for other types of intersecting line segments?
4. What is the significance of proving triangles are similar before congruent in this case?
5. How does angle congruence help in proving side relationships in triangles?

**Tip:** In geometry, always look for vertical angles or corresponding angles when dealing with intersecting lines—they are often congruent and crucial to proving triangle similarity or congruence.

Segments AD and BC intersect at point O. Prove that AC = BD, and m∠ABC = m∠BAD.

Learn how to prove that AC = BD and ∠ABC = ∠BAD in intersecting triangles using the AAA similarity theorem and vertical angles congruence. This problem is suitable for high school students studying geometry.

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