## Proving Congruence of Triangles ABD and CBD in Triangle ABC### Problem StatementGiven: - Triangle \( \triangle ABC \)- \( \overline{AB} \cong \overline{BC} \)- \( \overline{BD} \) bisects \( \angle ABC \)Prove: \( \triangle ABD \cong \triangle CBD \)### Diagram AnalysisThe 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 CongruenceTo 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 TheoremUsing 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 trianglesThus, by SAS Congruence Theorem, \( \triangle ABD \cong \triangle CBD \).### ConclusionWe have successfully proven that \( \triangle ABD \) is congruent to \( \triangle CBD \) given the initial conditions and the SAS Congruence Theorem.

Name: ## Proving Congruence of Triangles ABD and CBD in Triangle ABC### Pro
Uploaded: 2024-03-20T06:47:08.000Z
Channel: Bartleby
Description: ## Proving Congruence of Triangles ABD and CBD in Triangle ABC### Problem StatementGiven: - Triangle \( \triangle ABC \)- \( \overline{AB} \cong \overline{BC} \)- \( \overline{BD} \) bisects \( \angle ABC \)Prove: \( \triangle ABD \cong \triangle CB