Lesson 17A: The Side-Angle-Side (SAS) Two Triangles to be Similar
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1 : The Side-Angle-Side (SAS) Two Triangles to be Similar Learning Target 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. Original coordinates D (, ) R (, ) G (, ) D 2 r y axis R 90 ( DRG) Is this composition a similarity transformation?
2 Concepts to remember from lesson Two triangles KML and HJI are similar if there is a similarity transformation that maps KML and HJI. So KML~ HJI, the similarity transformation takes K to H, M to J, and L to I, 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? Given two triangles ABC and A B C so that A B = A C and AB AC m A = m A, then the triangles are similar, ABC ~ A B C. = = and m = m, than Example 1) Using the definition above, are the two triangles below similar? Explain your answer
3 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 P is common to both triangle PQR and triangle PQ R. We are also given information about some of the side lengths. 2. How can the information provided be used to determine whether PQR is similar to PQ R? 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 PQR and PQ R. What do you notice? The side lengths are not proportional. 4. Based on your work in parts (a) (c), draw a conclusion about the relationship between PQR and PQ R. Explain your reasoning.
4 : 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 = Yes, the triangles shown are similar. ABC ~ DEF by SAS because m B = m E, 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. ABC ~ ADE by AA because m ADE = m ABC, and both triangles share A.. 3. Are the triangles shown below similar? Explain. If the triangles are similar, write the similarity statement 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.
5 4. Given ABC and LMN in the diagram below, and B L, 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 AB ML = CB LN because the cross products of the proportion 8 3 = are both 38. We are given that 4 L B. Therefore, ABC~ MLN by the SAS 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 120 angle, and a second triangle has a 65 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 60, which means that it is not possible for the triangle to have a 65 angle. For the triangles to be similar, both triangles would have to have angles measuring 120 and 65, but this is impossible due to the angle sum of a triangle. 7. 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? 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.
6 : 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 B and D are the only similar triangles because they have the same angle measures. Using the angle sum of a triangle, each of the triangles B and D have angles of 75, 60, and For each part (a) through (c) below, state which of the three triangles, if any, are similar and why. Triangles A and B are similar because they have two pairs of corresponding sides that are in the same ratio and their included angles are equal measures. Triangle C cannot be shown similar because even though it has two sides that are the same length as two sides of triangle A, the 70 angle in triangle C is not the included angle and, therefore, does not correspond to the 70 angle in triangle A. 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 50, 60, and 70. Therefore, ABC~ TSR.
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3. SOLUTION: Since point T is on the x-axis, the y-coordinate of the point will be 0. On the triangle it is indicated that.
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