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9. Given: E is the midpoint of AB and CD Prove: Δ4ΕC ΔΒED

9. Given: E is the midpoint of AB and CD Prove: Δ4ΕC ΔΒED

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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### Problem Statement **Given:** E is the midpoint of line segments \( \overline{AB} \) and \( \overline{CD} \). **Prove:** \( \triangle AEC \cong \triangle BED \) ### Diagram Description The diagram shows two triangles, \( \triangle AEC \) and \( \triangle BED \), intersecting at point E. Point E is labeled as the midpoint on both line segments \( \overline{AB} \) and \( \overline{CD} \). ### Proof Structure The proof is organized in a two-column format with "Statements" on the left and "Reasons" on the right. There are five numbered steps provided for each section: #### Statements 1. E is the midpoint of \( \overline{AB} \) and \( \overline{CD} \). 2. \( \overline{AE} \cong \overline{EB} \) 3. \( \overline{CE} \cong \overline{ED} \) 4. \( \angle AEC \cong \angle BED \) 5. \( \triangle AEC \cong \triangle BED \) #### Reasons 1. Given 2. Definition of midpoint 3. Definition of midpoint 4. Vertical Angles Theorem 5. SAS Congruence Postulate This format helps in understanding the logical flow required to prove the congruency of the two triangles, \( \triangle AEC \) and \( \triangle BED \).
Transcribed Image Text:### Problem Statement **Given:** E is the midpoint of line segments \( \overline{AB} \) and \( \overline{CD} \). **Prove:** \( \triangle AEC \cong \triangle BED \) ### Diagram Description The diagram shows two triangles, \( \triangle AEC \) and \( \triangle BED \), intersecting at point E. Point E is labeled as the midpoint on both line segments \( \overline{AB} \) and \( \overline{CD} \). ### Proof Structure The proof is organized in a two-column format with "Statements" on the left and "Reasons" on the right. There are five numbered steps provided for each section: #### Statements 1. E is the midpoint of \( \overline{AB} \) and \( \overline{CD} \). 2. \( \overline{AE} \cong \overline{EB} \) 3. \( \overline{CE} \cong \overline{ED} \) 4. \( \angle AEC \cong \angle BED \) 5. \( \triangle AEC \cong \triangle BED \) #### Reasons 1. Given 2. Definition of midpoint 3. Definition of midpoint 4. Vertical Angles Theorem 5. SAS Congruence Postulate This format helps in understanding the logical flow required to prove the congruency of the two triangles, \( \triangle AEC \) and \( \triangle BED \).
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