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Lesson 5 COMMON CORE MATHEMATICS CURRICULUM U4 GEOMETRY Name__________________________________________ Period______ Date____________ Lesson 5: Triangle Similarity Criteria The Angle-Angle (AA) Criterion for Two Triangles to be Similar Classwork Exercises 1β5 1. Below are 3 triangles with two pairs of equal angles. Measure the lengths of the corresponding sides to verify that the ratio of their lengths is proportional. 1 Lesson 5 COMMON CORE MATHEMATICS CURRICULUM U4 GEOMETRY 2. Are the triangles similar? Explain. 3. Why is it that you only needed to construct triangles where two pairs of angles were equal and not three? 4. Why were the ratios of the corresponding sides proportional? 5. Do you think that what you observed will always be true when you construct a pair of triangles with two pairs of equal angles? Explain. Exercises 6β9 6. Are the triangles shown below similar? Explain. If the triangles are similar, identify any missing angle and side length measures. 28 20 3 8 6 44.42° 57.12° 7 78.46° 57.12° 2 COMMON CORE MATHEMATICS CURRICULUM Lesson 5 U4 GEOMETRY 7. Are the triangles shown below similar? Explain. If the triangles are similar identify any missing angle and side length measures. 8. The triangles shown below are similar. Use what you know about similar triangles to find the missing side lengths π₯ and π¦. 3 COMMON CORE MATHEMATICS CURRICULUM Lesson 5 U4 GEOMETRY 9. The triangles shown below are similar. Write an explanation to a student, Claudia, of how to find the lengths of π₯ and π¦. The Side-Angle-Side (SAS) and Side-Side-Side (SSS) Criteria for Two Triangles to be Similar Exploratory Challenge 1/Exercises 1β2 1. Examine the figure and answer the questions to determine whether or not the triangles shown are similar. Redraw the figure as two separate triangles before answering the questions. a. What information is given about the triangles in Figure 1? b. How can the information provided be used to determine whether β³ π΄π΅πΆ is similar to β³ π΄π΅β² πΆ β² ? 4 Lesson 5 COMMON CORE MATHEMATICS CURRICULUM U4 GEOMETRY 2. β² β² c. Compare the corresponding side lengths of β³ π΄π΅πΆ and β³ π΄π΅ πΆ . What do you notice? d. Based on your work in parts (a)β(c), draw a conclusion about the relationship between β³ π΄π΅πΆ and β³ π΄π΅β² πΆ β². Explain your reasoning. Examine the figure, and answer the questions to determine whether or not the triangles shown are similar. a. What information is given about the triangles in Figure 2? b. How can the information provided be used to determine whether β³ πππ is similar to β³ ππβ²π β²? 5 COMMON CORE MATHEMATICS CURRICULUM Lesson 5 U4 GEOMETRY c. Compare the corresponding side lengths of β³ πππ and β³ ππβ²π β². What do you notice? d. Based on your work in parts (a)β(c), draw a conclusion about the relationship between β³ πππ and β³ ππβ²π β². Explain your reasoning. Exploratory Challenge 2/Exercises 3β4 3. Examine the figure and answer the questions to determine whether or not the triangles shown are similar. a. What information is given about the triangles in Figure 3? b. How can the information provided be used to determine whether β³ π΄π΅πΆ is similar to β³ π΄π΅β² πΆ β²? c. Compare the corresponding side lengths of β³ π΄π΅πΆ and β³ π΄π΅β² πΆ β² . What do you notice? d. Based on your work in parts (a)β(c), make a conjecture about the relationship between β³ π΄π΅πΆ and β³ π΄π΅β² πΆ β² . Explain your reasoning. 6 COMMON CORE MATHEMATICS CURRICULUM Lesson 5 U4 GEOMETRY 4. Examine the figure and answer the questions to determine whether or not the triangles shown are similar. a. What information is given about the triangles in Figure 4? b. How can the information provided be used to determine whether β³ π΄π΅πΆ is similar to β³ π΄π΅β² πΆ β²? c. Compare the corresponding side lengths of β³ π΄π΅πΆ and β³ π΄π΅β² πΆ β². What do you notice? d. Based on your work in parts (a)β(c), make a conjecture about the relationship between β³ π΄π΅πΆ and β³ π΄π΅β² πΆ β². Explain your reasoning. 7 COMMON CORE MATHEMATICS CURRICULUM Lesson 5 U4 GEOMETRY Exercises 5β10 5. Are the triangles shown below similar? Explain. If the triangles are similar, write the similarity statement. 6. Are the triangles shown below similar? Explain. If the triangles are similar, write the similarity statement. 7. Are the triangles shown below similar? Explain. If the triangles are similar, write the similarity statement. 8 COMMON CORE MATHEMATICS CURRICULUM Lesson 5 U4 GEOMETRY 8. Are the triangles shown below similar? Explain. If the triangles are similar, write the similarity statement. 9. Are the triangles shown below similar? Explain. If the triangles are similar, write the similarity statement. 10. Are the triangles shown below similar? Explain. If the triangles are similar, write the similarity statement. 9 COMMON CORE MATHEMATICS CURRICULUM Lesson 5 U4 GEOMETRY 11. Given the diagram below, π½π» = 7.5, π»πΎ = 6, and πΎπΏ = 9, is there a pair of similar triangles? If so, write a similarity statement and explain why. If not, explain your reasoning. Conclusions about the similarity criterion for triangles. ο§ Given only information about the angles of a pair of triangles, how can you determine if the given triangles are similar? ο§ Given only information about one pair of angles for two triangles, how can you determine if the given triangles are similar? ο§ Given no information about the angles of a pair of triangles, how can you determine if the given triangles are similar? 10 COMMON CORE MATHEMATICS CURRICULUM Lesson 5 U4 GEOMETRY Problem Set - Homework 1. For each part (a) through (d) below, state which of the three triangles, if any, are similar and why. a. b. c. d. 11 COMMON CORE MATHEMATICS CURRICULUM Lesson 5 U4 GEOMETRY 2. For each given pair of triangles, determine if the triangles are similar or not, and provide your reasoning. If the triangles are similar, write a similarity statement relating the triangles. a. b. c. d. 12 COMMON CORE MATHEMATICS CURRICULUM Lesson 5 U4 GEOMETRY 3. For each pair of similar triangles below, determine the unknown lengths of the sides labeled with letters. a. b. 4. Given that Μ Μ Μ Μ π΄π· and Μ Μ Μ Μ π΅πΆ intersect at πΈ, and Μ Μ Μ Μ π΄π΅ β₯ Μ Μ Μ Μ πΆπ· , show that βπ΄π΅πΈ~βπ·πΆπΈ. 13 COMMON CORE MATHEMATICS CURRICULUM Lesson 5 U4 GEOMETRY 5. Given π΅πΈ = 11, πΈπ΄ = 11, π΅π· = 7, and π·πΆ = 7, show that βπ΅πΈπ·~βπ΅π΄πΆ. . 6. Given the diagram below, π is on Μ Μ Μ Μ π π and π is on Μ Μ Μ Μ π π , ππ = 2, ππ = 6, ππ = 9, and ππ = 4. a. Show that βπ ππ~βπ ππ. b. Find π π and π π. 7. One triangle has a 120° angle, and a second triangle has a 65° angle. Is it possible that the two triangles are similar? Explain why or why not. 8. A right triangle has a leg that is 12 cm long, and another right triangle has a leg that is 6 cm long. Are the two triangles similar or not? If so, explain why. If not, what other information would be needed to show they are similar? 14