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In this proof, we have the given information \( \overline{AC} \perp \overline{BD} \), meaning that line segment \( \overline{AC} \) is perpendicular to line segment \( \overline{BD} \). We are trying to prove that two triangles within this figure are congruent. Here’s a step-by-step breakdown to fill in the flowchart proof:

1. **Given Information:**
   - \( \overline{AC} \perp \overline{BD} \)
   - This establishes that \( \angle BDA \) and \( \angle BDC \) are both right angles (perpendicular lines form right angles).

2. **Identify Right Angles:**
   - **Statement:** \( \angle BDA \) and \( \angle BDC \) are right angles.
   - **Reason:** Definition of perpendicular lines.

3. **Congruent Angles (Right Angles):**
   - **Statement:** \( \angle BDA \cong \angle BDC \).
   - **Reason:** All right angles are congruent.

4. **Shared Side:**
   - **Statement:** \( \overline{BD} \) is congruent to \( \overline{BD} \).
   - **Reason:** Reflexive property of congruence (a segment is congruent to itself).

5. **Mark Equal Sides:**
   - Based on the diagram, it appears \( \overline{AB} \cong \overline{CB} \).
   - **Statement:** \( \overline{AB} \cong \overline{CB} \).
   - **Reason:** Given (as indicated in the diagram by matching tick marks).

6. **Apply Triangle Congruence:**
   - **Statement:** \( \triangle ABD \cong \triangle CBD \).
   - **Reason:** By the **Hypotenuse-Leg (HL) theorem** (since both triangles are right triangles and share a hypotenuse \( \overline{BD} \)).

Would you like more details, or do you have any specific questions about this proof?

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Here are five related questions for deeper understanding:

1. What other triangle congruence criteria could be applied if different information were given?
2. How would the proof change if \( \overline{AB} \neq \overline{CB} \)?
3. Why is the Reflexive Property essential in proofs like this?
4. Could the Angle-Side-Angle (ASA) criterion be used here?
5. How does knowing two angles are right angles simplify the proof?

**Tip:** Always look for shared sides or angles between triangles in proofs, as these often help establish congruence through reflexive properties.

Given AC ⊥ BD, complete the flowchart proof below. Note that the last statement and reason have both been filled in for you.

Learn how to complete a flowchart proof for triangle congruence, given AC ⊥ BD. This problem involves geometric concepts, including perpendicular lines, right angle congruence, and the Hypotenuse-Leg (HL) theorem. Ideal for students in Grades 9-10.

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