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In the given flowchart, we are tasked with completing the proof for the congruence of triangles \( \triangle ABC \cong \triangle ADC \) using the information that \( \overline{AC} \) bisects \( \angle BCD \). The last statement and reason in the proof have already been filled in, indicating that the triangles are congruent by Angle-Angle-Side (AAS).

Here is a step-by-step breakdown of how to complete the flowchart:

1. **First Box (Given):** 
   - Statement: \( \overline{AC} \text{ bisects } \angle BCD \)
   - Reason: Given.

2. **Second Box (Angle Bisector Property):**
   - Statement: \( \angle BCA \cong \angle DCA \)
   - Reason: An angle bisector divides an angle into two congruent angles.

3. **Third Box (Given Angle Congruence):**
   - Statement: \( \angle D \cong \angle B \)
   - Reason: Given.

4. **Fourth Box (Reflexive Property):**
   - Statement: \( \overline{AC} \cong \overline{AC} \)
   - Reason: Reflexive property of congruence (since the segment is shared by both triangles).

These pieces of information lead to the final statement:

5. **Conclusion (Congruent Triangles):**
   - Statement: \( \triangle ABC \cong \triangle ADC \)
   - Reason: AAS (Angle-Angle-Side).

This proof shows that triangles \( \triangle ABC \) and \( \triangle ADC \) are congruent based on the congruence of two angles and the included side.

Would you like any more detailed explanations or clarifications on this?

Here are five related questions to further expand on this:
1. What is the AAS (Angle-Angle-Side) congruence criterion?
2. How does the angle bisector theorem apply to this problem?
3. Why is the reflexive property of congruence important in proving triangles congruent?
4. Can you identify other triangle congruence criteria besides AAS?
5. How would this proof change if \( \overline{AC} \) did not bisect \( \angle BCD \)?

**Tip:** Always double-check which properties or theorems apply directly, especially when dealing with bisectors or parallel lines.

Given AC bisects ∠BCD, complete the flowchart proof below. What are the missing statements and reasons to show ΔABC ≅ ΔADC?

Solve a flowchart proof where AC bisects ∠BCD to show ΔABC ≅ ΔADC using the AAS congruence criterion. Explore the use of the Angle Bisector Theorem and the Reflexive Property of Congruence. Suitable for high school students in Grades 9-12.

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