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The pair of triangles shown are because the sides are and correspondihg angles are 9. 12 15 37 37

The pair of triangles shown are because the sides are and correspondihg angles are 9. 12 15 37 37

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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**Title: Understanding Triangle Similarity** **Introduction** In this section, we will discuss the concept of similar triangles. Two triangles are said to be similar if their corresponding angles are equal and the lengths of their corresponding sides are proportional. **Example Problem** Consider the two triangles shown in the diagram below: **Triangle 1 (△QRS):** - \(\angle R\) is a right angle (90°) - \(\angle QRS = 37°\) - Side \(QR\) = 12 units - Side \(RS\) = 9 units - Side \(QS\) = 15 units **Triangle 2 (△JKL):** - \(\angle J\) is a right angle (90°) - \(\angle KJL = 37°\) - Side \(KJ\) = 6 units - Side \(JL\) = 8 units - Side \(KL\) = 10 units **Analysis of the Triangles:** To determine if the two triangles are similar, let's check the following: 1. **Angle Correspondence:** - \(\angle R\) in △QRS is 90° and \(\angle J\) in △JKL is also 90°. Thus, \(\angle R = \angle J\). - \(\angle QRS\) in △QRS is 37° and \(\angle KJL\) in △JKL is also 37°. Thus, \(\angle QRS = \angle KJL\). 2. **Side Proportions:** - \(\frac{QR}{KJ} = \frac{12}{6} = 2\) - \(\frac{RS}{JL} = \frac{9}{4.5} = 2\) (Incorrect value to be taken \(\frac{9}{8}\)) - \(\frac{QS}{KL} = \frac{15}{10} = 1.5\) (Incorrect value to be taken \(\frac{15}{10}\)) **Observation:** The ratio of \(RS\) and \(JL\); \(QS\) and \(KL\) does not give a perfectly equal side ratio. 3. **Conclusion:** Since the corresponding angles are equal and the ratios of the corresponding sides are proportional, we can conclude that the pair of triangles
Transcribed Image Text:**Title: Understanding Triangle Similarity** **Introduction** In this section, we will discuss the concept of similar triangles. Two triangles are said to be similar if their corresponding angles are equal and the lengths of their corresponding sides are proportional. **Example Problem** Consider the two triangles shown in the diagram below: **Triangle 1 (△QRS):** - \(\angle R\) is a right angle (90°) - \(\angle QRS = 37°\) - Side \(QR\) = 12 units - Side \(RS\) = 9 units - Side \(QS\) = 15 units **Triangle 2 (△JKL):** - \(\angle J\) is a right angle (90°) - \(\angle KJL = 37°\) - Side \(KJ\) = 6 units - Side \(JL\) = 8 units - Side \(KL\) = 10 units **Analysis of the Triangles:** To determine if the two triangles are similar, let's check the following: 1. **Angle Correspondence:** - \(\angle R\) in △QRS is 90° and \(\angle J\) in △JKL is also 90°. Thus, \(\angle R = \angle J\). - \(\angle QRS\) in △QRS is 37° and \(\angle KJL\) in △JKL is also 37°. Thus, \(\angle QRS = \angle KJL\). 2. **Side Proportions:** - \(\frac{QR}{KJ} = \frac{12}{6} = 2\) - \(\frac{RS}{JL} = \frac{9}{4.5} = 2\) (Incorrect value to be taken \(\frac{9}{8}\)) - \(\frac{QS}{KL} = \frac{15}{10} = 1.5\) (Incorrect value to be taken \(\frac{15}{10}\)) **Observation:** The ratio of \(RS\) and \(JL\); \(QS\) and \(KL\) does not give a perfectly equal side ratio. 3. **Conclusion:** Since the corresponding angles are equal and the ratios of the corresponding sides are proportional, we can conclude that the pair of triangles
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