For A ABC, ĀB= BE and BD bisects LABC. Prove AABD =A CBD. し B

Algebra and Trigonometry (6th Edition)
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## Proving Congruence of Triangles ABD and CBD in Triangle ABC

### Problem Statement
Given: 
- Triangle \( \triangle ABC \)
- \( \overline{AB} \cong \overline{BC} \)
- \( \overline{BD} \) bisects \( \angle ABC \)

Prove: \( \triangle ABD \cong \triangle CBD \)

### Diagram Analysis
The diagram helps to visualize the given information and the proof. It shows:
- Triangle \( \triangle ABC \) with points A, B, and C.
- Line segment \( \overline{AB} \) is congruent to line segment \( \overline{BC} \).
- Point D lies on line segment \( \overline{AC} \).
- Line segment \( \overline{BD} \) bisects \( \angle ABC \).

### Steps to Prove Congruence
To prove \( \triangle ABD \) is congruent to \( \triangle CBD \), we will use the Side-Angle-Side (SAS) Congruence Theorem.

1. **Identify Congruent Line Segments:**
   - Given \( \overline{AB} \cong \overline{BC} \), we have one pair of congruent sides.

2. **Identify Angles:**
   - \( \overline{BD} \) bisects \( \angle ABC \), thus \( \angle ABD \cong \angle CBD \).

3. **Identify Congruent Line Segments:**
   - \( \overline{BD} \) is common to both \( \triangle ABD \) and \( \triangle CBD \).

### Applying Theorem
Using the SAS (Side-Angle-Side) Theorem:
- Side \( \overline{AB} \cong \overline{BC} \)
- Angle \( \angle ABD \cong \angle CBD \)
- Side \( \overline{BD} \) is common to both triangles

Thus, by SAS Congruence Theorem, \( \triangle ABD \cong \triangle CBD \).

### Conclusion
We have successfully proven that \( \triangle ABD \) is congruent to \( \triangle CBD \) given the initial conditions and the SAS Congruence Theorem.
Transcribed Image Text:## Proving Congruence of Triangles ABD and CBD in Triangle ABC ### Problem Statement Given: - Triangle \( \triangle ABC \) - \( \overline{AB} \cong \overline{BC} \) - \( \overline{BD} \) bisects \( \angle ABC \) Prove: \( \triangle ABD \cong \triangle CBD \) ### Diagram Analysis The diagram helps to visualize the given information and the proof. It shows: - Triangle \( \triangle ABC \) with points A, B, and C. - Line segment \( \overline{AB} \) is congruent to line segment \( \overline{BC} \). - Point D lies on line segment \( \overline{AC} \). - Line segment \( \overline{BD} \) bisects \( \angle ABC \). ### Steps to Prove Congruence To prove \( \triangle ABD \) is congruent to \( \triangle CBD \), we will use the Side-Angle-Side (SAS) Congruence Theorem. 1. **Identify Congruent Line Segments:** - Given \( \overline{AB} \cong \overline{BC} \), we have one pair of congruent sides. 2. **Identify Angles:** - \( \overline{BD} \) bisects \( \angle ABC \), thus \( \angle ABD \cong \angle CBD \). 3. **Identify Congruent Line Segments:** - \( \overline{BD} \) is common to both \( \triangle ABD \) and \( \triangle CBD \). ### Applying Theorem Using the SAS (Side-Angle-Side) Theorem: - Side \( \overline{AB} \cong \overline{BC} \) - Angle \( \angle ABD \cong \angle CBD \) - Side \( \overline{BD} \) is common to both triangles Thus, by SAS Congruence Theorem, \( \triangle ABD \cong \triangle CBD \). ### Conclusion We have successfully proven that \( \triangle ABD \) is congruent to \( \triangle CBD \) given the initial conditions and the SAS Congruence Theorem.
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