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Focus on Math Concepts Lesson 19 Part 1: Introduction CCSS 7.G.A.2 Understand Conditions for Drawing Triangles What side lengths make a triangle? You have worked with triangles for several years. You know they have three sides and three angles and they seem pretty simple. But, can a triangle have just any side lengths? Can a triangle have just any angle measures? Try an experiment. Pick random numbers, or roll a number cube and try drawing a triangle with those side lengths. You will find some combinations of side lengths make a triangle. In this lesson you will learn which combinations do or don’t work as sides of a triangle. Can these side lengths form a triangle? Think If you know its side lengths, you know everything about a triangle. Cut a straw into lengths 3 cm, 4 cm, and 5 cm. Use the three pieces to form a triangle, and carefully trace the triangle on paper. Then rearrange the sides to form another triangle and trace that one too. Compare your two triangles, and compare yours to other students’ triangles. Compare your triangles to the one shown here. Be sure to measure the angles with a protractor and compare the angles, too. Trace one of your triangles next to the one shown here. If you work carefully, you will be able to understand why we can say that if you know the side lengths you know “everything” about a triangle. Any two triangles with given side lengths always have the same angle measures, too. 180 L19: Understand Conditions for Drawing Triangles ©Curriculum Associates, LLC Copying is not permitted. Part 1: Introduction Lesson 19 Think If you know its angle measures, you don’t know everything about a triangle. Use a protractor to draw a triangle with one 908 angle, one 308 angle, and one 608 angle. Try to draw another triangle with the exact same angle measures, but with at least one side a different length. If you can do that, try another one. How do these triangles compare to yours? When you have drawn several triangles with angles measuring 308, 608, and 908, compare them. Compare your triangles to the two shown here, and to other students’ triangles. Reflect 1 In what way does knowing the measures of three sides of a triangle tell you more than just knowing the measures of the three angles? L19: Understand Conditions for Drawing Triangles ©Curriculum Associates, LLC Copying is not permitted. 181 Part 2: Guided Instruction Lesson 19 Explore It Explore each set of conditions described below. On a separate sheet of paper draw several triangles to match the description. Then make a conjecture about the number of different triangles possible, and justify your thinking. 2 Three sides are given: Use side lengths 6 cm, 6 cm, and 4 cm. How many different triangles can you make? 3 Three angles are given: Start with angle measures 808, 408, and 608. Try other sets of angles such as 208, 1208, and 408. Try three measures that don’t add up to 1808. 4 Two sides and the angle between them are given: Start with side lengths 5 cm and 4 cm, meeting to form a 458 angle. Then try side lengths of 2 cm and 4 cm, meeting to form a 90° angle. 5 Two angles and the side between them are given: Draw triangles with a side 4 cm long and a 45° angle at each endpoint. Try different angles measures and different lengths for the side between them. Now try this problem. 6 How many triangles can you draw with side lengths 2 cm, 3 cm, and 5 cm? Try sides 2 cm, 3 cm, and 8 cm. Explain what you find. 182 L19: Understand Conditions for Drawing Triangles ©Curriculum Associates, LLC Copying is not permitted. Part 2: Guided Instruction Lesson 19 Make Sense of It Solve the problems below as a group. 7 Use straws to form a quadrilateral with side lengths 5 cm, 5 cm, 3 cm, and 3 cm. Make a sketch and label the sides of your quadrilateral in the space to the right. 8 Now form different quadrilaterals using the same side lengths. Sketch and label the sides of at least two of the quadrilaterals you find. 9 Experiment with some other dimensions, such as 3 cm, 4 cm, 5 cm, and 6 cm, or 4 cm, 4 cm, 4 cm, and 4 cm. Can you find a set of 4 side lengths that will form only one quadrilateral? Explain why or why not. Try It Another Way Work with your group to understand why triangles are used in building. 10 When you have the three side lengths of a triangle, there is only one triangle that can be formed. How does that explain why a gate like the one here is built with a diagonal board? L19: Understand Conditions for Drawing Triangles ©Curriculum Associates, LLC Copying is not permitted. 183 Part 3: Guided Practice Lesson 19 Connect It Talk through these problems as a class, then write your answers below. 11 Analyze: Draw an isosceles triangle with exactly one 408 angle. Is this the only possibility or can you draw another triangle that will also meet these conditions? How is this different from drawing a triangle given 2 sides and the angle between them? 40° 12 Explain: How many different triangles can be drawn with two obtuse angles? Explain how you can be sure of your answer without drawing many examples. 13 Create: Sketch several triangles with one side 6 cm long, one side 10 cm long, and the third side x cm long. What are some possible lengths of the third side? Write an inequality to express the least possible value of x. Write another inequality to express the greatest possible value of x. Use a sketch to explain your thinking. 184 L19: Understand Conditions for Drawing Triangles ©Curriculum Associates, LLC Copying is not permitted. Part 4: Common Core Performance Task Lesson 19 Put It Together 14 Use what you have learned in this lesson to complete this task. A Give conditions (measurements) for a triangle that would result in more than one possible triangle. Include at least one side length. Explain your thinking and provide an illustration. B Give conditions for a triangle that would result in only one possible triangle. Explain your thinking and provide an illustration. L19: Understand Conditions for Drawing Triangles ©Curriculum Associates, LLC Copying is not permitted. 185