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Given: BD 1 AC, BD bisects ZABC Prove: AABD == ACBD A D

Given: BD 1 AC, BD bisects ZABC Prove: AABD == ACBD A D

Elementary Geometry For College Students, 7e
Elementary Geometry For College Students, 7e
7th Edition
ISBN:9781337614085
Author:Alexander, Daniel C.; Koeberlein, Geralyn M.
Publisher:Alexander, Daniel C.; Koeberlein, Geralyn M.
ChapterP: Preliminary Concepts
SectionP.CT: Test
Problem 1CT
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**Transcription of Geometric Proof** **Given:** \( BD \perp AC, \, BD \) bisects \( \angle ABC \). **Prove:** \( \triangle ABD \cong \triangle CBD \). **Diagram Explanation:** The diagram illustrates triangle \( ABC \) with line \( BD \) drawn from point \( B \) to intersect at point \( D \) on line \( AC \). \( BD \) is perpendicular to \( AC \) and bisects \( \angle ABC \). **Proof Structure:** | **Statements** | **Reasons** | |------------------------------------------|------------------------------------------| | 1) \( BD \perp AC, \, BD \) bisects \( \angle ABC \). | 1) Given | | 2) \( \angle ADB = \angle CDB \) | 2) Definition of perpendicular | | 3) \( \angle ABD = \angle CBD \) | 3) Definition of angle bisector | | 4) \( BD = BD \) | 4) Reflexive Property of \(\cong\) | | 5) \( \triangle ABD \cong \triangle CBD \) | 5) ASA (Angle-Side-Angle) Postulate | This proof demonstrates that triangles \( \triangle ABD \) and \( \triangle CBD \) are congruent through the ASA postulate, using the given information and properties of angles and perpendicular bisectors.
Transcribed Image Text:**Transcription of Geometric Proof** **Given:** \( BD \perp AC, \, BD \) bisects \( \angle ABC \). **Prove:** \( \triangle ABD \cong \triangle CBD \). **Diagram Explanation:** The diagram illustrates triangle \( ABC \) with line \( BD \) drawn from point \( B \) to intersect at point \( D \) on line \( AC \). \( BD \) is perpendicular to \( AC \) and bisects \( \angle ABC \). **Proof Structure:** | **Statements** | **Reasons** | |------------------------------------------|------------------------------------------| | 1) \( BD \perp AC, \, BD \) bisects \( \angle ABC \). | 1) Given | | 2) \( \angle ADB = \angle CDB \) | 2) Definition of perpendicular | | 3) \( \angle ABD = \angle CBD \) | 3) Definition of angle bisector | | 4) \( BD = BD \) | 4) Reflexive Property of \(\cong\) | | 5) \( \triangle ABD \cong \triangle CBD \) | 5) ASA (Angle-Side-Angle) Postulate | This proof demonstrates that triangles \( \triangle ABD \) and \( \triangle CBD \) are congruent through the ASA postulate, using the given information and properties of angles and perpendicular bisectors.
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