Similar Triangles or Not?
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- Sydney Blaise Shields
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1 Similar Triangles or Not? Similar Triangle Theorems.2 Learning Goals In this lesson, you will: Use constructions to explore similar triangle theorems. xplore the Angle-Angle (AA) Similarity Theorem. xplore the Side-Side-Side (SSS) Similarity Theorem. xplore the Side-Angle-Side (SAS) Similarity Theorem. Key Terms Angle-Angle Similarity Theorem Side-Side-Side Similarity Theorem included angle included side Side-Angle-Side Similarity Theorem An art projector is a piece of equipment that artists have used to create exact copies of artwork, to enlarge artwork, or to reduce artwork. A basic art projector uses a light bulb and a lens within a box. The light rays from the art being copied are collected onto a lens at a single point. The lens then projects the image of the art onto a screen as shown. If the projector is set up properly, the triangles shown will be similar polygons. You can show that these triangles are similar without measuring all of the side lengths and all of the interior angles. 451
2 Problem 1 Using Two Angles In the previous lesson, you used transformations to prove that triangles are similar when their corresponding angles are congruent and their corresponding sides are proportional. In this problem, you will explore the similarity of two triangles using construction tools. 1. Identify all of the corresponding congruent angles and all of the corresponding proportional sides using the similar triangles shown. RST WXY R S W X Y T You can conclude that two triangles are similar if you are able to prove that all of their corresponding angles are congruent and all of their corresponding sides are proportional. Let s use constructions to see if you can use fewer pairs of angles or fewer pairs of sides to show that triangles are similar. 2. Construct triangle using only and in triangle as shown. Make all the corresponding side lengths of triangle different from the side lengths of triangle. 452 Chapter Similarity Through Transformations
3 3. Measure the angles and sides of triangle and triangle. Are the two triangles similar? xplain your reasoning. id everyone construct the same triangle? 4. In triangles and, two pairs of corresponding angles are congruent. etermine if this is sufficient information to conclude that the triangles are similar. The Angle-Angle Similarity Theorem states: If two angles of one triangle are congruent to two angles of another triangle, then the triangles are similar. B A C If m A 5 m and m C 5 m, then ABC. 5. xplain why this similarity theorem is Angle-Angle instead of Angle-Angle-Angle..2 Similar Triangle Theorems 453
4 . The triangles shown are isosceles triangles. o you have enough information to show that the triangles are similar? xplain your reasoning. M Q L N P R 7. The triangles shown are isosceles triangles. o you have enough information to show that the triangles are similar? xplain your reasoning. T W S U V X 454 Chapter Similarity Through Transformations
5 Problem 2 Using Two and Three Proportional Sides 1. Construct triangle by doubling the lengths of sides and. Construct the new and separately and then construct the triangle. This will ensure a ratio of 2 : 1. o not duplicate angles. 2. Measure the angles and sides of triangle and triangle. Are the two triangles similar? xplain your reasoning. id everyone construct the same triangle? 3. Two pairs of corresponding sides are proportional. etermine if this is sufficient information to conclude that the triangles are similar. Not having sufficient information doesn t mean that the triangle are NOT similar. It just means that you can t know for sure whether the triangles are or are not similar..2 Similar Triangle Theorems 455
6 4. Construct triangle by doubling the lengths of sides,, and. Construct the new side lengths separately, and then construct the triangle. o not duplicate angles. What about now? id everyone construct the same triangle? 5. Measure the angles and sides of triangle and triangle. Are the two triangles similar? xplain your reasoning.. Three pairs of corresponding sides are proportional. etermine if this is sufficient information to conclude that the triangles are similar. 45 Chapter Similarity Through Transformations
7 The Side-Side-Side Similarity Theorem states: If all three corresponding sides of two triangles are proportional, then the triangles are similar. B A C If AB 5 BC 5 AC, then ABC.? Stacy says that the Side-Side-Side Similarity Theorem tells us that two triangles can have proportional sides, but not congruent angles, and still be similar. Michael doesn t think that s right, but he can t explain why. 7. Is Stacy correct? If not, explain why not. 8. etermine whether UVW is similar to XYZ. If so, use symbols to write a similarity statement. 33 meters V 24 meters 1 meters Y W 22 meters Z 24 meters 3 meters X U 9. escribe how transformations could be used to determine whether two triangles are similar when all pairs of corresponding sides are proportional..2 Similar Triangle Theorems 457
8 Problem 3 Using Two Proportional Sides and an Angle An included angle is an angle formed by two consecutive sides of a figure. An included side is a line segment between two consecutive angles of a figure. 1. Construct triangle by duplicating an angle and doubling the length of the two sides that make up that angle. Construct the new side lengths separately, and then construct the triangle. 2. Measure the angles and sides of triangle and triangle. Are the two triangles similar? xplain your reasoning. 458 Chapter Similarity Through Transformations
9 3. Two pairs of corresponding sides are proportional and the corresponding included angles are congruent. etermine if this is sufficient information to conclude that the triangles are similar. 4. escribe how transformations could be used to determine whether two triangles are similar when two pairs of corresponding sides are proportional and the included angles are congruent. The Side-Angle-Side Similarity Theorem states: If two of the corresponding sides of two triangles are proportional and the included angles are congruent, then the triangles are similar. B A C If AB 5 AC and A, then ABC..2 Similar Triangle Theorems 459
10 Talk the Talk 1. Gaelin is thinking of a triangle and he wants everyone in his class to draw a similar triangle. Complete the graphic organizer to describe the sides and angles of triangles he could provide. Be prepared to share your solutions and methods. 40 Chapter Similarity Through Transformations
11 Angle-Angle (AA) Side-Angle-Side (SAS) Similar Triangles Side-Side-Side (SSS) Combinations of Sides and Angles that o Not nsure Similarity.2 Similar Triangle Theorems 41
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