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.

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Answered: H Given: KGLJ: FK = HL Prove: FGHJ is a…

Math

Geometry

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

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.

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