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H Given: KGLJ: FK = HL Prove: FGHJ is a . K F G.

H Given: KGLJ: FK = HL Prove: FGHJ is a . K F G.

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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This is a geometry question.
**Question 18:** **Given:** Rectangle \( KGLJ \); \( FK = HL \) **Prove:** \( FGHI \) is a rectangle. **Diagram Description:** The diagram shows a rectangle \( FGHJ \) with points \( K \) and \( L \) on sides \( FG \) and \( HJ \) respectively. Point \( O \) is the intersection point of the diagonals \( KJ \) and \( LH \). The diagram illustrates several line segments within the rectangle, connecting various points to each other, creating smaller geometric shapes within the rectangle. **Explanation:** To prove that \( FGHJ \) is a rectangle, the following properties should be demonstrated: 1. Opposite sides are equal and parallel. 2. All angles are right angles. Given the properties of a rectangle and the conditions \( FK = HL \), one can conclude that specific symmetry properties hold, making \( FGHJ \) consistent with the properties of any rectangle. By stepping through standard geometric proofs and congruence relationships, it can be demonstrated that \( FGHJ \) maintains equal opposite sides, equal diagonals, and right angles. This exercise highlights the importance of understanding key properties and relationships within geometric shapes and applying theorem-based proofs to verify characteristics of composite shapes.
Transcribed Image Text:**Question 18:** **Given:** Rectangle \( KGLJ \); \( FK = HL \) **Prove:** \( FGHI \) is a rectangle. **Diagram Description:** The diagram shows a rectangle \( FGHJ \) with points \( K \) and \( L \) on sides \( FG \) and \( HJ \) respectively. Point \( O \) is the intersection point of the diagonals \( KJ \) and \( LH \). The diagram illustrates several line segments within the rectangle, connecting various points to each other, creating smaller geometric shapes within the rectangle. **Explanation:** To prove that \( FGHJ \) is a rectangle, the following properties should be demonstrated: 1. Opposite sides are equal and parallel. 2. All angles are right angles. Given the properties of a rectangle and the conditions \( FK = HL \), one can conclude that specific symmetry properties hold, making \( FGHJ \) consistent with the properties of any rectangle. By stepping through standard geometric proofs and congruence relationships, it can be demonstrated that \( FGHJ \) maintains equal opposite sides, equal diagonals, and right angles. This exercise highlights the importance of understanding key properties and relationships within geometric shapes and applying theorem-based proofs to verify characteristics of composite shapes.
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