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Triangle IJK is formed by connecting the midpoints of the side of triangle FGH. The measures of the interior angles of triangle FGH are shown. Find the measure of ZGIJ. Figures not necessarily drawn to scale. H 46 K J 60° F 74° G I

Triangle IJK is formed by connecting the midpoints of the side of triangle FGH. The measures of the interior angles of triangle FGH are shown. Find the measure of ZGIJ. Figures not necessarily drawn to scale. H 46 K J 60° F 74° G I

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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### Understanding Midpoints and Interior Angles in Triangles **Problem Statement:** Triangle \(IJK\) is formed by connecting the midpoints of the sides of triangle \(FGH\). The measures of the interior angles of triangle \(FGH\) are shown. Find the measure of \(\angle GIJ\). Figures not necessarily drawn to scale. **Diagram Details:** The provided diagram is a geometric illustration of triangle \(FGH\) with the following given angles: - \(\angle F = 60^\circ\) - \(\angle G = 74^\circ\) - \(\angle H = 46^\circ\) Within triangle \(FGH\), another triangle \(IJK\) is formed by connecting the midpoints of the sides of triangle \(FGH\). Points \(I\), \(J\), and \(K\) are the midpoints of the sides \(FG\), \(GH\), and \(HF\) respectively. The setup indicates that each side of triangle \(FGH\) has been bisected. **Diagram Explanation:** 1. **Main Triangle \(FGH\):** - Vertices: \(F\), \(G\), \(H\) - Given interior angles: \(\angle F (60^\circ)\), \(\angle G (74^\circ)\), \(\angle H (46^\circ)\) 2. **Secondary Triangle \(IJK\):** - Formed by joining the midpoints of the sides of triangle \(FGH\) - Points: \(I\), \(J\), \(K\) represent these midpoints **Objective:** The goal is to find the measure of \(\angle GIJ\). This involves understanding the properties of the angles and the relationships within the triangles, particularly involving midpoints and angle bisectors. **Geometric Insight:** Since \(IJK\) is formed by connecting the midpoints of \(FGH\), triangle \(IJK\) is similar to triangle \(FGH\) but each side of \(IJK\) is parallel to a side of \(FGH\) and half its length. The angles within triangle \(IJK\) correspond to the angles of triangle \(FGH\), a known property of mid-segment triangles in geometry. Using these properties and the given angles, students can deduce the measure of \(\angle GI
Transcribed Image Text:### Understanding Midpoints and Interior Angles in Triangles **Problem Statement:** Triangle \(IJK\) is formed by connecting the midpoints of the sides of triangle \(FGH\). The measures of the interior angles of triangle \(FGH\) are shown. Find the measure of \(\angle GIJ\). Figures not necessarily drawn to scale. **Diagram Details:** The provided diagram is a geometric illustration of triangle \(FGH\) with the following given angles: - \(\angle F = 60^\circ\) - \(\angle G = 74^\circ\) - \(\angle H = 46^\circ\) Within triangle \(FGH\), another triangle \(IJK\) is formed by connecting the midpoints of the sides of triangle \(FGH\). Points \(I\), \(J\), and \(K\) are the midpoints of the sides \(FG\), \(GH\), and \(HF\) respectively. The setup indicates that each side of triangle \(FGH\) has been bisected. **Diagram Explanation:** 1. **Main Triangle \(FGH\):** - Vertices: \(F\), \(G\), \(H\) - Given interior angles: \(\angle F (60^\circ)\), \(\angle G (74^\circ)\), \(\angle H (46^\circ)\) 2. **Secondary Triangle \(IJK\):** - Formed by joining the midpoints of the sides of triangle \(FGH\) - Points: \(I\), \(J\), \(K\) represent these midpoints **Objective:** The goal is to find the measure of \(\angle GIJ\). This involves understanding the properties of the angles and the relationships within the triangles, particularly involving midpoints and angle bisectors. **Geometric Insight:** Since \(IJK\) is formed by connecting the midpoints of \(FGH\), triangle \(IJK\) is similar to triangle \(FGH\) but each side of \(IJK\) is parallel to a side of \(FGH\) and half its length. The angles within triangle \(IJK\) correspond to the angles of triangle \(FGH\), a known property of mid-segment triangles in geometry. Using these properties and the given angles, students can deduce the measure of \(\angle GI
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