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This proof is a basic triangle congruence proof where you need to prove that triangles \(\triangle ABD \cong \triangle CBD\). Let's break it down step by step to complete the proof.

### Given Information:
1. \(AC \perp BD\) (so \( \angle ADB = 90^\circ\))
2. \( \angle A \cong \angle C \)

### Proof Steps So Far:
1. **Given:** \(AC \perp BD\) and \( \angle A \cong \angle C \)
2. **Reflexive Property:** \(BD \cong BD\) (This is the same side for both triangles)

### Completing the Proof:
Now we need a third step to show that \(\triangle ABD \cong \triangle CBD\).

### Step 3: Right Angle
Since \(AC \perp BD\), we know that both \(\angle ADB\) and \(\angle CDB\) are right angles. Thus:
\[
\angle ADB \cong \angle CDB
\]
**Reason:** Right angles are congruent.

### Step 4: Congruence
By the **AAS (Angle-Angle-Side)** Congruence Theorem (two angles and the non-included side), \(\triangle ABD \cong \triangle CBD\).

Thus, the completed proof would look like this:

| Step | Statement                               | Reason                        |
|------|-----------------------------------------|-------------------------------|
| 1    | \(AC \perp BD\), \( \angle A \cong \angle C \) | Given                         |
| 2    | \(BD \cong BD\)                         | Reflexive Property             |
| 3    | \(\angle ADB \cong \angle CDB\)         | Right angles are congruent     |
| 4    | \(\triangle ABD \cong \triangle CBD\)   | AAS Congruence Theorem         |

Let me know if you'd like more details or further clarification!

### 5 Related Questions:
1. What is the AAS Congruence Theorem, and how is it different from SAS?
2. How do you identify the reflexive property in a triangle proof?
3. What other congruence theorems can be used to prove triangle congruence?
4. How do perpendicular lines help in proving right angles in geometric proofs?
5. What is the significance of the given \( \angle A \cong \angle C \) in the proof?

### Tip:
When proving triangles are congruent, always look for common sides (reflexive property) and right angles if perpendicular lines are involved.

Given: AC ⊥ BD and ∠A ≅ ∠C. Prove: ΔABD ≅ ΔCBD.

Learn how to prove that triangles ΔABD and ΔCBD are congruent using the AAS Congruence Theorem. This geometric proof involves concepts like perpendicular lines, reflexive property, and right angles. Suitable for students in Grades 9-10.

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