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Given: E is the midpoint of AB and CD Prove: ΔΑEC - ΔΒED C

Given: E is the midpoint of AB and CD Prove: ΔΑEC - ΔΒED C

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 \( \overline{AB} \) and \( \overline{CD} \) **Prove:** \( \triangle AEC \cong \triangle BED \) **Diagram Description:** The diagram shows two triangles, \( \triangle AEC \) and \( \triangle BED \), with points A, B, C, D, and E. Point E is where diagonals \( \overline{AB} \) and \( \overline{CD} \) intersect. **Proof Structure:** | Statements | Reasons | |----------------------------------------------|-----------------------------| | E is the midpoint of \( \overline{AB} \) and \( \overline{CD} \) | Given | | \( \overline{AE} \cong \overline{BE} \) | Definition of Midpoint | | \( \overline{CE} \cong \overline{DE} \) | Definition of Midpoint | | \( \angle AEC \cong \angle BED \) | Vertical Angles Theorem | | \( \triangle AEC \cong \triangle BED \) | SAS (Side-Angle-Side) | **Notes on Proof:** 1. **Definitions:** - Midpoint: A point on a line segment that divides it into two equal parts. 2. **Theorems:** - Vertical Angles Theorem: If two angles are vertical angles, then they are congruent. 3. **Congruence Criteria:** - SAS (Side-Angle-Side): Two triangles are congruent if two sides and the included angle of one triangle are congruent to two sides and the included angle of another triangle.
Transcribed Image Text:**Problem Statement:** **Given:** E is the midpoint of \( \overline{AB} \) and \( \overline{CD} \) **Prove:** \( \triangle AEC \cong \triangle BED \) **Diagram Description:** The diagram shows two triangles, \( \triangle AEC \) and \( \triangle BED \), with points A, B, C, D, and E. Point E is where diagonals \( \overline{AB} \) and \( \overline{CD} \) intersect. **Proof Structure:** | Statements | Reasons | |----------------------------------------------|-----------------------------| | E is the midpoint of \( \overline{AB} \) and \( \overline{CD} \) | Given | | \( \overline{AE} \cong \overline{BE} \) | Definition of Midpoint | | \( \overline{CE} \cong \overline{DE} \) | Definition of Midpoint | | \( \angle AEC \cong \angle BED \) | Vertical Angles Theorem | | \( \triangle AEC \cong \triangle BED \) | SAS (Side-Angle-Side) | **Notes on Proof:** 1. **Definitions:** - Midpoint: A point on a line segment that divides it into two equal parts. 2. **Theorems:** - Vertical Angles Theorem: If two angles are vertical angles, then they are congruent. 3. **Congruence Criteria:** - SAS (Side-Angle-Side): Two triangles are congruent if two sides and the included angle of one triangle are congruent to two sides and the included angle of another triangle.
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