Algebra and Trigonometry (6th Edition)
6th Edition
ISBN:9780134463216
Author:Robert F. Blitzer
Publisher:Robert F. Blitzer
ChapterP: Prerequisites: Fundamental Concepts Of Algebra
Section: Chapter Questions
Problem 1MCCP: In Exercises 1-25, simplify the given expression or perform the indicated operation (and simplify,...
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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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