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4. If E is the midpoint of DF and CD = FG, then CE EG. C D E F G H

4. If E is the midpoint of DF and CD = FG, then CE EG. C D E F G H

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:** 4. If \( E \) is the midpoint of \( \overline{DF} \) and \( \overline{CD} \cong \overline{FG} \), then \( \overline{CE} \cong \overline{EG} \). **Diagram Explanation:** The diagram shows a rectangle labeled with points \( C, D, E, F, G \) along the top side and a point \( H \) at the bottom. Lines are drawn from points \( C, D, E, F, \) and \( G \) to \( H \). The segments within the rectangle visually support the geometric relationships described in the problem. **Reasoning:** 1. **Given:** The conditions \( E \) is the midpoint of \( \overline{DF} \) and \( \overline{CD} \cong \overline{FG} \) are provided. 2. **Def midpoint:** As \( E \) is the midpoint of \( \overline{DF} \), by definition, \( \overline{DE} \cong \overline{EF} \). 3. **Def \(\cong\) seg:** Since \( \overline{CD} \cong \overline{FG} \), segments \( CD \) and \( FG \) are congruent. 4. **Segment Addition Postulate (Seg Add Post):** According to the postulate, the whole segment is equal to the sum of its parts; therefore, \( \overline{CE} = \overline{CD} + \overline{DE} \) and \( \overline{EG} = \overline{EF} + \overline{FG} \). 5. **Substitution (Subst):** Since \( \overline{DE} \cong \overline{EF} \) and \( \overline{CD} \cong \overline{FG} \), substituting gives \( \overline{CE} = \overline{CD} + \overline{DE} = \overline{EF} + \overline{FG} = \overline{EG} \). 6. **Conclusion (Def \(\cong\) seg):** Hence, \( \overline{CE} \cong \overline{EG} \). This logical sequence
Transcribed Image Text:**Problem Statement:** 4. If \( E \) is the midpoint of \( \overline{DF} \) and \( \overline{CD} \cong \overline{FG} \), then \( \overline{CE} \cong \overline{EG} \). **Diagram Explanation:** The diagram shows a rectangle labeled with points \( C, D, E, F, G \) along the top side and a point \( H \) at the bottom. Lines are drawn from points \( C, D, E, F, \) and \( G \) to \( H \). The segments within the rectangle visually support the geometric relationships described in the problem. **Reasoning:** 1. **Given:** The conditions \( E \) is the midpoint of \( \overline{DF} \) and \( \overline{CD} \cong \overline{FG} \) are provided. 2. **Def midpoint:** As \( E \) is the midpoint of \( \overline{DF} \), by definition, \( \overline{DE} \cong \overline{EF} \). 3. **Def \(\cong\) seg:** Since \( \overline{CD} \cong \overline{FG} \), segments \( CD \) and \( FG \) are congruent. 4. **Segment Addition Postulate (Seg Add Post):** According to the postulate, the whole segment is equal to the sum of its parts; therefore, \( \overline{CE} = \overline{CD} + \overline{DE} \) and \( \overline{EG} = \overline{EF} + \overline{FG} \). 5. **Substitution (Subst):** Since \( \overline{DE} \cong \overline{EF} \) and \( \overline{CD} \cong \overline{FG} \), substituting gives \( \overline{CE} = \overline{CD} + \overline{DE} = \overline{EF} + \overline{FG} = \overline{EG} \). 6. **Conclusion (Def \(\cong\) seg):** Hence, \( \overline{CE} \cong \overline{EG} \). This logical sequence
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