Elementary Geometry For College Students, 7e
Elementary Geometry For College Students, 7e
7th Edition
ISBN: 9781337614085
Author: Alexander, Daniel C.; Koeberlein, Geralyn M.
Publisher: Cengage,
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Justify the reasons for the proof.

**Title:** Justify the Reasons for the Proof **Given:** - \( \overline{GE} \) bisects \( \angle DEF \) - \( \angle D \cong \angle F \) **Prove:** - \( \triangle DGE \cong \triangle FGE \) **Diagram Overview:** - The diagram features triangle \(DEF\) with point \(G\) on line \(EF\), creating two separate triangles, \(DGE\) and \(FGE\). Line \(GE\) is shown as an angle bisector of \( \angle DEF \). **Proof Structure:** | **Statement** | **Reason** | |--------------------------------------------|--------------------------------------| | 1. \( \overline{GE} \) bisects \( \angle DEF \) | 1. Given | | 2. \( \angle D \cong \angle F \) | 2. Given | | 3. \( \angle DEG \cong \angle FEG \) | 3. | | 4. \( \overline{GE} \cong \overline{GE} \) | 4. | | 5. \( \triangle DGE \cong \triangle FGE \) | 5. | **Reasons Key:** - Symmetric property - Reflexive property - Transitive property - Definition of midpoint - Definition of angle bisector - ASA (Angle-Side-Angle) - SSS (Side-Side-Side) - AAS (Angle-Angle-Side) - SAS (Side-Angle-Side) Each reason must be justified using the properties or definitions listed in the key.
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Transcribed Image Text:**Title:** Justify the Reasons for the Proof **Given:** - \( \overline{GE} \) bisects \( \angle DEF \) - \( \angle D \cong \angle F \) **Prove:** - \( \triangle DGE \cong \triangle FGE \) **Diagram Overview:** - The diagram features triangle \(DEF\) with point \(G\) on line \(EF\), creating two separate triangles, \(DGE\) and \(FGE\). Line \(GE\) is shown as an angle bisector of \( \angle DEF \). **Proof Structure:** | **Statement** | **Reason** | |--------------------------------------------|--------------------------------------| | 1. \( \overline{GE} \) bisects \( \angle DEF \) | 1. Given | | 2. \( \angle D \cong \angle F \) | 2. Given | | 3. \( \angle DEG \cong \angle FEG \) | 3. | | 4. \( \overline{GE} \cong \overline{GE} \) | 4. | | 5. \( \triangle DGE \cong \triangle FGE \) | 5. | **Reasons Key:** - Symmetric property - Reflexive property - Transitive property - Definition of midpoint - Definition of angle bisector - ASA (Angle-Side-Angle) - SSS (Side-Side-Side) - AAS (Angle-Angle-Side) - SAS (Side-Angle-Side) Each reason must be justified using the properties or definitions listed in the key.
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