5.2. In mathematics, when a geometric figure is transformed, the size and shape of the. Hey, Haven t I Seen You Before? Congruent Triangles
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1 Hey, Haven t I Seen You Before? Congruent Triangles. Learning Goals In this lesson, you will: Identify corresponding sides and corresponding angles of congruent triangles. Explore the relationship between the corresponding sides of congruent triangles. Explore the relationship between the corresponding angles of congruent triangles. Write congruence statements for congruent triangles. Identify and use rigid motion to create new images. In mathematics, when a geometric figure is transformed, the size and shape of the figure do not change. However, in physics, things are a little different. An idea known as length contraction explains that when an object is in motion, its length appears to be slightly less than it really is. This cannot be seen with everyday objects because they do not move fast enough. To truly see this phenomenon you would have to view an object moving close to the speed of light. In fact, if an object was moving past you at the speed of light, the length of the object would seem to be practically zero! This theory is very difficult to prove and yet scientists came up with the idea in the late 100s. How do you think scientists test and prove length contraction? Do you think geometry is used in these verifications? 37
2 Problem 1 Understanding Congruence 1. Graph triangle ABC by plotting the points A (, 10), B (1, ), and C (, ). y Let s explore the properties of congruent triangles. 0 x a. Classify triangle ABC. Explain your reasoning. b. Calculate the length of side AB. I would use the Pythagorean Theorem to calculate the length.. Translate triangle ABC 10 units to the left to form triangle DEF. Graph triangle DEF and list the coordinates of points D, E, and F. 3 Chapter Congruence Through Transformations
3 Triangle ABC and triangle DEF in Question 1 are the same size and the same shape. Each side of triangle ABC matches, or corresponds to, a specific side of triangle DEF. 3. Given what you know about corresponding sides of congruent triangles, predict the side lengths of triangle DEF. Corresponding sides are sides that have the same relative positions in corresponding geometric figures.. Verify your prediction. a. Identify the pairs of corresponding sides of triangle ABC and triangle DEF. b. Determine the side lengths of triangle DEF. c. Compare the lengths of the sides of triangle ABC to the lengths of the corresponding sides of triangle DEF. What do you notice?. In general, what can you conclude about the relationship between the corresponding sides of congruent triangles? Would there ever be a time when corresponding sides of figures would not be congruent?. Congruent Triangles 39
4 Use triangle ABC and triangle DEF from Question 1 to answer each question.. Use a protractor to determine the measures of /A, /B, and /C. Each angle in triangle ABC corresponds to a specific angle in triangle DEF. Corresponding angles are angles that have the same relative positions in corresponding geometric figures. 7. What would you predict to be true about the measures of corresponding angles of congruent triangles?. Verify your prediction. a. Identify the corresponding angles of triangle ABC and triangle DEF. b. Use a protractor to determine the measures of angles D, E, and F. c. Compare the measures of the angles of triangle ABC to the measures of the corresponding angles of triangle DEF. 9. In general, what can you conclude about the relationship between the corresponding angles of congruent triangles? So, what can you say about corresponding sides and corresponding angles of congruent triangles? 30 Chapter Congruence Through Transformations
5 Problem Statements of Triangle Congruence 1. Consider the congruence statement njrb > nmns. a. Identify the congruent angles. b. Identify the congruent sides.. Analyze the two triangles shown. Remember, the > means is congruent to. y (1, ) T W (9, ) P (, 1) C (, 3) 0 (3, ) K x (, 9) M a. Determine the transformation used to create triangle PMK. b. Does the transformation preserve the size and shape of the triangle in this problem situation? Why or why not? c. Write a triangle congruence statement for the triangles. d. Identify the congruent angles and congruent sides.. Congruent Triangles 31
6 3. Analyze the two triangles shown. y T (, 9) 0 G (3, ) R (, 1) Q (, 1) x V (3, ) Z (, 9) a. Determine the transformation used to create triangle ZQV. b. Does the transformation preserve the size and shape of the triangle in this problem situation? Why or why not? c. Write a triangle congruence statement for the triangles shown. d. Identify the congruent angles. e. Identify the congruent sides. 3 Chapter Congruence Through Transformations
7 Talk the Talk 1. Given any triangle on a coordinate plane, how can you create a different triangle that you know will be congruent to the original triangle?. Describe the properties of congruent triangles. Be prepared to share your solutions and methods.. Congruent Triangles 33
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