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
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
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 are not necessarily drawn to scale.
### Diagram Description
#### Given Information:
- Triangle \( FGH \) with interior angles \( \angle F = 60^\circ \), \( \angle H = 46^\circ \), and \( \angle G = 74^\circ \).
- A smaller triangle \( IJK \) is formed inside \( FGH \) by connecting the midpoints of its sides.
#### Diagram Analysis:
1. The larger triangle \( FGH \) forms the outer boundary.
2. Triangle \( IJK \), a smaller triangle, is inscribed within \( FGH \) by joining the midpoints of the sides of \( FGH \).
3. \(\triangle IJK\) is smaller and similar to \( \triangle FGH \) due to the midsegment theorem, which states that the segment joining the midpoints of two sides of a triangle is parallel to the third side and is half its length.
4. Since midpoints are connected, \( IJK \) should have parallel sides to \(FGH\) forming similar but scaled-down angles.
### Solution Objective:
Find the value of \( \angle GIJ \), an interior angle of the smaller triangle \( IJK \).
In a transformed and similar figure like this, each angle in \( IJK \) corresponds to the angles in the parent triangle. Therefore, the angle \( \angle GIJ \) we seek in \( IJK \) corresponds to one of the original angles of \(\triangle FGH\), maintaining the properties of similarity and equal angles. Hence, \( \angle GIJ \) equals \( 60^\circ \), \( 46^\circ\), or \( 74^\circ\) as in \(\triangle FGH\).
### Answer:
The measure of \( \angle GIJ \) is equal to the angle at vertex \( G \) from \(\triangle FGH\), which is \( \angle G = 74^\circ\). Therefore:
\[ \angle GIJ = 74^\circ \]](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F6e1bdb0e-93e0-4643-a548-ab4acb8278f7%2Fa540ede9-27fb-490f-be34-3645e4eeb28d%2Fi74a3l_processed.png&w=3840&q=75)
Transcribed Image Text:### 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 are not necessarily drawn to scale.
### Diagram Description
#### Given Information:
- Triangle \( FGH \) with interior angles \( \angle F = 60^\circ \), \( \angle H = 46^\circ \), and \( \angle G = 74^\circ \).
- A smaller triangle \( IJK \) is formed inside \( FGH \) by connecting the midpoints of its sides.
#### Diagram Analysis:
1. The larger triangle \( FGH \) forms the outer boundary.
2. Triangle \( IJK \), a smaller triangle, is inscribed within \( FGH \) by joining the midpoints of the sides of \( FGH \).
3. \(\triangle IJK\) is smaller and similar to \( \triangle FGH \) due to the midsegment theorem, which states that the segment joining the midpoints of two sides of a triangle is parallel to the third side and is half its length.
4. Since midpoints are connected, \( IJK \) should have parallel sides to \(FGH\) forming similar but scaled-down angles.
### Solution Objective:
Find the value of \( \angle GIJ \), an interior angle of the smaller triangle \( IJK \).
In a transformed and similar figure like this, each angle in \( IJK \) corresponds to the angles in the parent triangle. Therefore, the angle \( \angle GIJ \) we seek in \( IJK \) corresponds to one of the original angles of \(\triangle FGH\), maintaining the properties of similarity and equal angles. Hence, \( \angle GIJ \) equals \( 60^\circ \), \( 46^\circ\), or \( 74^\circ\) as in \(\triangle FGH\).
### Answer:
The measure of \( \angle GIJ \) is equal to the angle at vertex \( G \) from \(\triangle FGH\), which is \( \angle G = 74^\circ\). Therefore:
\[ \angle GIJ = 74^\circ \]
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