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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Question
In the given image, there are two triangles with labeled sides and angles. Your task is to determine which angle is larger between ∠1 and ∠2.
**Triangle 1:**
- Side opposite ∠1: 37
- Adjacent sides: 48 and 56
**Triangle 2:**
- Side opposite ∠2: 37
- Adjacent sides: 47 and 56
From the information given, we can use the Law of Cosines to compare the angles.
**Law of Cosines:**
c² = a² + b² - 2ab * cos(γ)
Applying the Law of Cosines to both triangles, we find that:
For Triangle 1:
56² = 37² + 48² - 2 * 37 * 48 * cos(∠1)
For Triangle 2:
56² = 37² + 47² - 2 * 37 * 47 * cos(∠2)
By solving these equations, you can find the cosines of each angle and determine which angle is larger. The larger angle will have a smaller cosine value.
This exercise is a practical example of using trigonometric identities to compare angles in triangles. Solving such problems helps in understanding the relationships between the sides and angles in geometrical figures.
**Graphical Explanation:**
The provided image contains two triangles with sides labeled. In Triangle 1, ∠1 is formed between the sides of length 37 and 56, while in Triangle 2, ∠2 is formed between the sides of length 37 and 56. By comparing the lengths of the other two sides in each triangle, we can establish that the greater the length of the side opposite the given angle (while keeping the other sides constant), the greater the angle.
In this specific case, since the side 48 in Triangle 1 is longer than the side 47 in Triangle 2, and both have a side of 37 and 56, it can be concluded that ∠1 is larger than ∠2.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F20047395-754f-4b4f-b55f-051358973abe%2Fc60305d2-d4ff-4148-bd2a-7935cdda8ead%2Fl8bl8t8.jpeg&w=3840&q=75)
Transcribed Image Text:**Which angle is larger, ∠1 or ∠2?**
In the given image, there are two triangles with labeled sides and angles. Your task is to determine which angle is larger between ∠1 and ∠2.
**Triangle 1:**
- Side opposite ∠1: 37
- Adjacent sides: 48 and 56
**Triangle 2:**
- Side opposite ∠2: 37
- Adjacent sides: 47 and 56
From the information given, we can use the Law of Cosines to compare the angles.
**Law of Cosines:**
c² = a² + b² - 2ab * cos(γ)
Applying the Law of Cosines to both triangles, we find that:
For Triangle 1:
56² = 37² + 48² - 2 * 37 * 48 * cos(∠1)
For Triangle 2:
56² = 37² + 47² - 2 * 37 * 47 * cos(∠2)
By solving these equations, you can find the cosines of each angle and determine which angle is larger. The larger angle will have a smaller cosine value.
This exercise is a practical example of using trigonometric identities to compare angles in triangles. Solving such problems helps in understanding the relationships between the sides and angles in geometrical figures.
**Graphical Explanation:**
The provided image contains two triangles with sides labeled. In Triangle 1, ∠1 is formed between the sides of length 37 and 56, while in Triangle 2, ∠2 is formed between the sides of length 37 and 56. By comparing the lengths of the other two sides in each triangle, we can establish that the greater the length of the side opposite the given angle (while keeping the other sides constant), the greater the angle.
In this specific case, since the side 48 in Triangle 1 is longer than the side 47 in Triangle 2, and both have a side of 37 and 56, it can be concluded that ∠1 is larger than ∠2.
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