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Lesson 17A NYS COMMON CORE MATHEMATICS CURRICULUM Name:__________________________________ M2 Period: ________ Date: __________ Lesson 17A: The Side-Angle-Side (SAS) Two Triangles to be Similar Learning Target ο· I can use the side-angle-side criterion for two triangles to be similar to solve triangle problems. Opening exercise State the coordinates of the image of the following composition of transformations. π·2 β ππ¦βππ₯ππ β π 90° (βπ·π πΊ) Original coordinates π·( , ) π ( , ) πΊ( , ) Is this composition a similarity transformation? Lesson 17A NYS COMMON CORE MATHEMATICS CURRICULUM Name:__________________________________ M2 Period: ________ Date: __________ Concepts to remember from lesson 14-15 Two triangles β³ πΎππΏ and β³ π»π½πΌ are similar if there is a similarity transformation that maps β³ πΎππΏ and β³ π»π½πΌ . So β³ πΎππΏ~ β³ π»π½πΌ, the similarity transformation takes πΎ to π», π to π½, and πΏ to πΌ, such that the corresponding angles are equal in measurement and the corresponding lengths of sides are proportional. Also we learned AA similarity criteria - Two triangles can be considered similar if they have two pairs of corresponding equal angles. A New Condition for Similarity: S-A-S Similarity Two triangles are ______________________ if they have one pair of _____________ ____________that are congruent and the sides adjacent to that angle are proportional This is called the ____________ ______ _____________ criterion. How do we prove that two tringles are similar? π΄β² π΅β² π΄β² πΆ β² Given two triangles β³ π΄π΅πΆ and β³ π΄βπ΅βπΆβ so that = and π΄π΅ π΄πΆ β² β² β² β² πβ π΄ = πβ π΄ , then the triangles are similar, β³ π΄π΅πΆ ~ β³ π΄ π΅ πΆ . = = and πβ = πβ , than Example 1) Using the definition above, are the two triangles below similar? Explain your answer Lesson 17A NYS COMMON CORE MATHEMATICS CURRICULUM Name:__________________________________ M2 Period: ________ Date: __________ Example 2) Use the figure at right to answer the following questions. a) Name and state all the triangles that you see in Figure 1. Draw and label each triangle separately in the space below. b) Are the corresponding sides of these two triangles proportional? Show the work that leads to your answer. c) Are the included angles of these two triangles congruent? Give a reason for your answer. d) Are the two triangles similar? If so, write a similarity statement. Example 3 Examine the figure, and answer the questions to determine whether or not the triangles shown are similar. 1. What information is given about the triangles in Figure 2? Ans: We are given that β π· is common to both triangle β³ π·πΈπΉ and triangle β³ π·πΈβ²πΉβ². We are also given information about some of the side lengths. 2. How can the information provided be used to determine whether β³ π·πΈπΉ is similar to β³ π·πΈβ²πΉβ²? Ans: We know that similar triangles will have ratios of corresponding sides that are proportional; therefore, we can use the side lengths to check for proportionality. 3. Compare the corresponding side lengths of β³ π·πΈπΉ and β³ π·πΈβ²πΉβ². What do you notice? π ππ β ππ β ππ π π The side lengths are not proportional. 4. Based on your work in parts (a)β(c), draw a conclusion about the relationship between β³ π·πΈπΉ and β³ π·πΈβ²πΉβ². Explain your reasoning. Lesson 17A NYS COMMON CORE MATHEMATICS CURRICULUM Name:__________________________________ M2 Period: ________ Date: __________ Lesson 17A: The Side-Angle-Side (SAS) Two Triangles to be Similar Classwork 1. Are the triangles shown below similar? Explain. If the triangles are similar, write the similarity statement. 3.13 3 = 1.565 1.5 Yes, the triangles shown are similar. β³ π΄π΅πΆ ~ β³ π·πΈπΉ by SAS because mβ π΅ = mβ πΈ, and the adjacent sides are proportional. 2. Are the triangles shown below similar? Explain. If the triangles are similar, write the similarity statement. Yes, the triangles shown are similar. β³ π¨π©πͺ ~ β³ π¨π«π¬ by AA because π¦β π¨π«π¬ = π¦β π¨π©πͺ, and both triangles share β π¨. . 3. Are the triangles shown below similar? Explain. If the triangles are similar, write the similarity statement. π π β π π There is no information about the angle measures other than the right angle, so we cannot use AA to conclude the triangles are similar. We only have information about two of the three side lengths for each triangle, so we cannot use SSS to conclude they are similar. If the triangles are similar, we would have to use the SAS criterion, and since the side lengths are not proportional, the triangles shown are not similar. Lesson 17A NYS COMMON CORE MATHEMATICS CURRICULUM Name:__________________________________ M2 Period: ________ Date: __________ 4. Given βπ΄π΅πΆ and βπΏππ in the diagram below, and β π΅ β β πΏ, determine if the triangles are similar. If so, write a similarity statement, and state the criterion used to support your claim. In comparing the ratios of sides between figures, I found that the cross products of the proportion π π = π ππ π π π π π¨π© π΄π³ = πͺπ© π³π΅ because are both ππ. We are given that β π³ β β π©. Therefore, βπ¨π©πͺ~βπ΄π³π΅ by the πΊπ¨πΊ criterion for proving similar triangles. 5. For each pair of similar triangles below, determine the unknown lengths of the sides labeled with letters. a. b. . 6. One triangle has a πππ° angle, and a second triangle has a ππ° angle. Is it possible that the two triangles are similar? Explain why or why not. No, the triangles cannot be similar because in the first triangle, the sum of the remaining angles is ππ°, which means that it is not possible for the triangle to have a ππ° angle. For the triangles to be similar, both triangles would have to have angles measuring πππ° and ππ°, but this is impossible due to the angle sum of a triangle. 7. A right triangle has a leg that is ππ ππ¦ long, and another right triangle has a leg that is π ππ¦ 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? The two triangles may or may not be similar. There is not enough information to make this claim. If the second leg of the first triangle is twice the length of the second leg of the first triangle, then the triangles are similar by SAS criterion for showing similar triangles. Lesson 17A NYS COMMON CORE MATHEMATICS CURRICULUM Name:__________________________________ M2 Period: ________ Date: __________ Lesson 17A: The Side-Angle-Side (SAS) Two Triangles to be Similar Homework - Problem Set 1. For each part (a) through (d) below, state which of the three triangles, if any, are similar and why. Triangles π© and π« are the only similar triangles because they have the same angle measures. Using the angle sum of a triangle, each of the triangles π© and π« have angles of ππ°, ππ°, and ππ°. 2. For each part (a) through (c) below, state which of the three triangles, if any, are similar and why. Triangles π¨ and π© are similar because they have two pairs of corresponding sides that are in the same ratio and their included angles are equal measures. Triangle πͺ cannot be shown similar because even though it has two sides that are the same length as two sides of triangle π¨, the ππ° angle in triangle πͺ is not the included angle and, therefore, does not correspond to the ππ° angle in triangle π¨. 3. 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. The triangles are similar because, using the angle sum of a triangle, each triangle has angle measures of ππ°, ππ°, and ππ°. Therefore, βπ¨π©πͺ~βπ»πΊπΉ.