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Given: GJ bisects FH, and FH bisects GJ. Prove: ΔFGK < ΔΗK

Given: GJ bisects FH, and FH bisects GJ. Prove: ΔFGK < ΔΗK

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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### Given: \(\overline{GJ}\) bisects \(\overline{FH}\), and \(\overline{FH}\) bisects \(\overline{GJ}\). ### Prove: \(\triangle FGK \cong \triangle HJK\) ### Diagram Explanation: A geometric diagram is shown on the right side which consists of two intersecting lines, labeled \(\overline{FH}\) and \(\overline{GJ}\). The intersection point of these two lines is labeled \(K\). - The endpoints of one line are marked as \(F\) and \(H\). - The endpoints of the other line are labeled \(G\) and \(J\). ### Conjecture: Based on the given information, we need to prove that the triangles \( \triangle FGK \) and \( \triangle HJK \) are congruent to each other. ### Approach: To prove the congruence of the two triangles: 1. Utilize the fact that \( \overline{GJ} \) and \( \overline{FH} \) are bisected at their intersection point \( K \). 2. Use geometrical properties and congruence theorems such as Side-Angle-Side (SAS) or Angle-Side-Angle (ASA) to show the required congruence. This problem involves applying the concepts of line segment bisectors and triangle congruence to arrive at the solution.
Transcribed Image Text:### Given: \(\overline{GJ}\) bisects \(\overline{FH}\), and \(\overline{FH}\) bisects \(\overline{GJ}\). ### Prove: \(\triangle FGK \cong \triangle HJK\) ### Diagram Explanation: A geometric diagram is shown on the right side which consists of two intersecting lines, labeled \(\overline{FH}\) and \(\overline{GJ}\). The intersection point of these two lines is labeled \(K\). - The endpoints of one line are marked as \(F\) and \(H\). - The endpoints of the other line are labeled \(G\) and \(J\). ### Conjecture: Based on the given information, we need to prove that the triangles \( \triangle FGK \) and \( \triangle HJK \) are congruent to each other. ### Approach: To prove the congruence of the two triangles: 1. Utilize the fact that \( \overline{GJ} \) and \( \overline{FH} \) are bisected at their intersection point \( K \). 2. Use geometrical properties and congruence theorems such as Side-Angle-Side (SAS) or Angle-Side-Angle (ASA) to show the required congruence. This problem involves applying the concepts of line segment bisectors and triangle congruence to arrive at the solution.
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