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Gven AG HJ~AKLM, which must be true? Select all that apply a A GJ = KM B. HJ=LM C GH %3D KL KM D. m2J = mZM %3D m/K 23

Gven AG HJ~AKLM, which must be true? Select all that apply a A GJ = KM B. HJ=LM C GH %3D KL KM D. m2J = mZM %3D m/K 23

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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### Analysis of Triangle Similarity #### Incorrect Given \(\triangle GHJ \sim \triangle KLM\), which must be true? Select all that apply. - **A.** \(GJ = KM\) - **B.** \(HJ = LM\) - **C.** \(\frac{GH}{KL} = \frac{GJ}{KM}\) - **D.** \(m\angle J = m\angle M\) - **E.** \(\frac{m\angle G}{m\angle K} = \frac{m\angle H}{m\angle L}\) ### Explanation: In the scenario above, we are given that two triangles, \(\triangle GHJ\) and \(\triangle KLM\), are similar (\(\triangle GHJ \sim \triangle KLM\)). Triangle similarity implies that all corresponding angles are equal and the lengths of corresponding sides are proportional. Below, we will evaluate each statement to determine which must be true based on the properties of similar triangles: - **A.** \(GJ = KM\) This suggests that the lengths of sides \(GJ\) and \(KM\) are equal. However, for similar triangles, the corresponding sides are proportional, not necessarily equal unless the similarity ratio is 1:1. - **B.** \(HJ = LM\) This suggests that the lengths of sides \(HJ\) and \(LM\) are equal. Similar to option A, this is not necessarily true for similar triangles unless the ratio is 1:1. - **C.** \(\frac{GH}{KL} = \frac{GJ}{KM}\) This is true because, by the definition of similar triangles, the ratios of the lengths of corresponding sides are equal. If \(\triangle GHJ \sim \triangle KLM\), then \(\frac{GH}{KL} = \frac{HJ}{LM} = \frac{GJ}{KM}\). - **D.** \(m\angle J = m\angle M\) This is true because corresponding angles in similar triangles are equal. If \(\triangle GHJ \sim \triangle KLM\), then \(m\angle G = m\angle K\), \(m\angle H = m\angle L\), and \(m\angle J = m\angle M\). - **E
Transcribed Image Text:### Analysis of Triangle Similarity #### Incorrect Given \(\triangle GHJ \sim \triangle KLM\), which must be true? Select all that apply. - **A.** \(GJ = KM\) - **B.** \(HJ = LM\) - **C.** \(\frac{GH}{KL} = \frac{GJ}{KM}\) - **D.** \(m\angle J = m\angle M\) - **E.** \(\frac{m\angle G}{m\angle K} = \frac{m\angle H}{m\angle L}\) ### Explanation: In the scenario above, we are given that two triangles, \(\triangle GHJ\) and \(\triangle KLM\), are similar (\(\triangle GHJ \sim \triangle KLM\)). Triangle similarity implies that all corresponding angles are equal and the lengths of corresponding sides are proportional. Below, we will evaluate each statement to determine which must be true based on the properties of similar triangles: - **A.** \(GJ = KM\) This suggests that the lengths of sides \(GJ\) and \(KM\) are equal. However, for similar triangles, the corresponding sides are proportional, not necessarily equal unless the similarity ratio is 1:1. - **B.** \(HJ = LM\) This suggests that the lengths of sides \(HJ\) and \(LM\) are equal. Similar to option A, this is not necessarily true for similar triangles unless the ratio is 1:1. - **C.** \(\frac{GH}{KL} = \frac{GJ}{KM}\) This is true because, by the definition of similar triangles, the ratios of the lengths of corresponding sides are equal. If \(\triangle GHJ \sim \triangle KLM\), then \(\frac{GH}{KL} = \frac{HJ}{LM} = \frac{GJ}{KM}\). - **D.** \(m\angle J = m\angle M\) This is true because corresponding angles in similar triangles are equal. If \(\triangle GHJ \sim \triangle KLM\), then \(m\angle G = m\angle K\), \(m\angle H = m\angle L\), and \(m\angle J = m\angle M\). - **E
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