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DG4TE_883_04.qxd 10/24/06 6:38 PM Page 215 LESSON YO LESSON 4.3 4.3 Triangle Inequalities How long must each side of this drawbridge be so that the bridge spans the river when both sides come down? PLANNING Readers are plentiful, thinkers are rare. LESSON OUTLINE HARRIET MARTINEAU One day: 25 min Investigation 10 min Sharing 5 min Closing 5 min Exercises MATERIALS Drawbridges over the Chicago River in Chicago, Illinois The sum of the lengths of the two parts of the drawbridge must be equal to or greater than the distance across the waterway. Triangles have similar requirements. You can build a triangle using one blue rod, one green rod, and one red rod. Could you still build a triangle if you used a yellow rod instead of the green rod? Why or why not? Could you form a triangle with two yellow rods and one green rod? What if you used two green rods and one yellow rod? construction tools protractors sticks or uncooked spaghetti, optional Triangle Exterior Angle Conjecture (W), optional Sketchpad activity Triangle Inequalities, optional TEACHING This lesson concerns three properties of triangles: the triangle inequality, the side-angle inequality, and the exterior angle property. How can you determine which sets of three rods can be arranged into triangles and which can’t? How do the measures of the angles in the triangle relate to the lengths of the rods? How is measure of the exterior angle formed by the yellow and blue rods in the triangle above related to the measures of the angles inside the triangle? In this lesson you will investigate these questions. LESSONSTANDARDS NCTM OBJECTIVES LESSON OBJECTIVES CONTENT Create a histogram and a stem-and-leaf PROCESSplot of a data set Given a list of data, use a calculator Problem to graphSolving a histogram Number histograms and stem-and-leaf plots Interpret Reasoning Algebra Decide the appropriateness of a histogram and a stem-andleafGeometry plot for a given data set Communication Measurement Connections Data/Probability Representation Investigate inequalities among sides and angles in triangles Discover the Exterior Angle Conjecture Practice construction skills Develop reasoning skills One Step To combine the investigations, pose this problem: “Draw a horizontal line, which we’ll call the base, and mark two points on it fairly close together. Make a triangle that has those two points as two of its vertices. Now, move one of the two points along the base, keeping the opposite side of the triangle fixed in length. Stop the movement at various points and measure all angles and side lengths. Look for patterns and make conjectures.” Students might use geometry software for their experimentation, or you might provide sticks or uncooked spaghetti. While circulating, be sure some groups focus on exterior angles, some on relative side and angle measures, and some on what happens when the triangle disappears. LESSON 4.3 Triangle Inequalities 215 DG4TE_883_04.qxd 10/24/06 6:38 PM INTRODUCTION To help students decide which rods form a triangle, you might have them copy to paper, or imagine copying, three different rods with the longest rod horizontal and the shorter rods trying to form the other sides of the triangle. [Ask] “Under what conditions can the shorter sides actually make a triangle?” Page 216 Investigation 1 What Is the Shortest Path from A to B? You will need ● ● a compass a straightedge Each person in your group should do each construction. Compare results when you finish. Step 1 Given: A T C T Construct: CAT Given: F S H S H F Step 2 The two shorter and FH , are sides, SH Step 2 not long enough to form a triangle with the . longest side, FS Construct: FSH You should have been able to construct CAT, but not FSH. Why? Discuss your results with others. State your observations as your next conjecture. C-20 Triangle Inequality Conjecture ? the length of the The sum of the lengths of any two sides of a triangle is greater than third side. The Triangle Inequality Conjecture relates the lengths of the three sides of a triangle. You can also think of it in another way: The shortest path between two points is along the segment connecting them. In other words, the path from A to C to B can’t be shorter than the path from A to B. Students can use the Dynamic Geometry Exploration at www.keymath.com/DG to help them understand the Triangle Inequality Conjecture. AB = 3 cm AC + CB = 4 cm C A keymath.com/DG 216 A C Guiding Investigation 1 Step 1 The construction can be done with either a compass or patty paper. You might want to have sticks or uncooked spaghetti available to represent the line segments. [Alert] Students may have difficulty seeing what the constructions demonstrate about the lengths of the three sides. [Ask] “What would happen if the lengths of the two smaller sides added up to exactly the same length as the third side or added up to less than that of the third side?” Don’t press students yet to understand in depth the connection to shortest paths; save that for Sharing. Construct a triangle with each set of segments as sides. CHAPTER 4 Discovering and Proving Triangle Properties B AB = 3 cm AC + CB = 3.1 cm AB = 3 cm AC + CB = 3.5 cm C A C B A B A AB = 3 cm AC + CB = 3 cm C B [ For an interactive version of this sketch, see the Dynamic Geometry Exploration The Triangle Inequality at www.keymath.com/DG .] DG4TE_883_04.qxd 10/24/06 6:38 PM Page 217 Investigation 2 Guiding Investigation 2 Where Are the Largest and Smallest Angles? Each person should draw a different scalene triangle for this investigation. Some group members should draw acute triangles, and some should draw obtuse triangles. You will need ● ● a ruler a protractor Measure the angles in your triangle. Label the angle with greatest measure L, the angle with second greatest measure M, and the smallest angle S. Step 1 M S M Measure the three sides. Label the longest side l, the second longest side m, and the shortest side s. Step 2 Step 3 The largest Step 3 side will be opposite the largest angle, and so on. L L S Which side is opposite L? M? S? Discuss your results with others. Write a conjecture that states where the largest and smallest angles are in a triangle, in relation to the longest and shortest sides. C-21 Side-Angle Inequality Conjecture In a triangle, if one side is longer than another side, then the angle opposite ? . larger than the angle opposite the shorter side the longer side is Adjacent interior angle Exterior angle Remote interior angles So far in this chapter, you have studied interior angles of triangles. Triangles also have exterior angles. If you extend one side of a triangle beyond its vertex, then you have constructed an exterior angle at that vertex. Each exterior angle of a triangle has an adjacent interior angle and a pair of remote interior angles. The remote interior angles are the two angles in the triangle that do not share a vertex with the exterior angle. Investigation 3 Exterior Angles of a Triangle Each person should draw a different scalene triangle for this investigation. Some group members should draw acute triangles, and some should draw obtuse triangles. You will need ● ● a straightedge patty paper Step 1 On your paper, draw a scalene triangle, ABC. beyond point B and label a point D Extend AB . Label the angles as outside the triangle on AB shown. Sharing Ideas (continued) [Ask] “Can anything be said about the difference in length of two sides of a triangle?” [The difference must be less than the length of the third side.] “Why must the difference be less than the length of the third side?” [Encourage students to use algebra to write the claims and see how one follows from the other: If a b c, then c b a.] [Alert] If students are confused by the symbols (greater than) and (less than), review their meaning. C c a A b x B D You might also use letters and inequality symbols in stating the second conjecture: If a b, then mA mB. The same investigation could also lead to the converse: If mA mB, then a b. When phrasing the third conjecture, introduce the terms adjacent, remote, interior, and exterior. [ELL] Adjacent means “next to”; remote means “far away”; interior means “on the inside”; exterior means “on the outside.” The wording of this conjecture may show a lot of variation among the groups. Encourage variety and creativity. Groups may want to reword more than the end of the conjecture. For example: “In a triangle, the smallest angle is opposite the shortest side and the largest angle is opposite the longest side.” Guiding Investigation 3 This investigation may be done as a follow-along activity. [Ask] “How could you state a Triangle Exterior Angle Inequality Conjecture?” [The measure of the exterior angle of a triangle must be greater than the measure of either remote interior angle.] SHARING IDEAS Have students present a variety of statements of conjectures. Lead the class in critiquing them and in reaching consensus about which conjecture to use. When you’ve reached consensus on the first conjecture, ask how to explain why the sum of the lengths of two sides of a triangle must be greater than the length of the third. Students will probably keep restating the fact that you just can’t make a triangle without that property. Ask about the relevance of the fact that the shortest path between two points is the line segment connecting them. Help students see that the shortest-path condition implies the Triangle Inequality Conjecture. [Ask] “Does the Triangle Inequality Conjecture imply the shortest-path condition?” [It doesn’t, because it makes no claim that other curves connecting the two points are longer than the line segment.] LESSON 4.3 Triangle Inequalities 217 DG4TE_883_04.qxd 10/24/06 6:38 PM Page 218 Assessing Progress You can assess students’ skill in constructing triangles, copying segments and angles, and measuring angles. Check their understanding of various kinds of triangles. Step 2 Copy the two remote interior angles, A and C, onto patty paper to show their sum. Step 3 How does the sum of a and c compare with x? Use your patty paper from Step 2 to compare. Step 4 Discuss your results with your group. State your observations as a conjecture. c a is equal to the sum of C-22 the measures of the remote ? . interior angles The measure of an exterior angle of a triangle Triangle Exterior Angle Conjecture Closing the Lesson Restate the three conjectures of this lesson: the Triangle Inequality Conjecture, the Side-Angle Inequality Conjecture, and the Triangle Exterior Angle Conjecture. If you plan to assign Exercise 11, also mention that the Triangle Inequality Conjecture implies that the difference in length of two sides of a triangle is less than the length of the third side; represent this inequality using letters and inequality symbols. If many students seem to be having difficulty, you may want to model sample problems, such as Exercise 5. BUILDING UNDERSTANDING Developing Proof The investigation may have convinced you that the Triangle Exterior Angle Conjecture is true, but can you explain why it is true for every triangle? As a group, discuss how to prove the Triangle Exterior Angle Conjecture. Use reasoning strategies such as draw a labeled diagram, represent a situation algebraically, and apply previous conjectures. Start by making a diagram and listing the relationships you already know among the angles in the diagram, then plan out the logic of your proof. You will write the paragraph proof of the Triangle Exterior Angle Conjecture in Exercise 17. EXERCISES In Exercises 1–4, determine whether it is possible to draw a triangle with sides having the given measures. If possible, write yes. If not possible, write no and make a sketch 3. 4 5 5 6 demonstrating why it is not possible. 2. 1. 3 cm, 4 cm, 5 cm yes 9 4. 3.5 cm, 4.5 cm, 7 cm yes In Exercises 5–10, use your new conjectures to arrange the unknown measures in order from greatest to least. a, b, c 5. 6. The exercises apply the three conjectures of this lesson. Ask students which conjectures they are using to solve each exercise. 3. 5 ft, 6 ft, 12 ft no 7. c, b, a c b c 2–18 evens 15 Portfolio 14 Journal 17 Group 1–15 odds Review 19–24 Algebra review 17 | Helping with the Exercises 70 8. a 17 in. 68 a a 9. a, c, b a, b, c w 10. 34 b b c z 72 c 30 a y 28 v 11. If 54 and 48 are the lengths of two sides of a triangle, what is the range of possible values for the length of the third side? 6 length 102 Exercise 11 As needed, remind students of the claims discussed in class: that the sum of the lengths of the two given sides is greater than the length of the third side, which is greater than the difference in the length of the two given sides. Exercises 1–4 If students are unsure, encourage them to make sketches. 218 b, a, c 12 cm 35 a 28 in. Performance assessment c 5 cm 15 in. Essential 9 cm b 55 b ASSIGNING HOMEWORK 12 2. 4 m, 5 m, 9 m no CHAPTER 4 Discovering and Proving Triangle Properties x 42 30 v, z, y, w, x DG4TE_883_04.qxd 10/24/06 6:38 PM Page 219 12. Developing Proof What’s wrong with this picture? Explain. By the Triangle Inequality Conjecture, the sum of 11 cm and 25 cm 25 cm should be greater than 48 cm. 11 cm 13. Developing Proof What’s wrong with this picture? Explain. 130 b 48 cm a In Exercises 14–16, use one of your new conjectures to find the missing measures. ? 14. t p ? 72° 15. r 135° p 135 125 b 55°, but 55° 130° 180°, which is impossible by the Triangle Sum Conjecture. 17. a b c 180° and x c 180°. Subtract c from both sides of both equations to get x 180 c and a b 180 c. Substitute a b for 180 c in the first equation to get x a b. ? 72° 16. x r t 130 x 58 144 17. Developing Proof Use the Triangle Sum Conjecture to explain why the Triangle Exterior Angle Conjecture is true. Use the figure at right. B b 18. Read the Recreation Connection below. If you want to know the perpendicular distance from a landmark to the path of your boat, what should be the measurement of your bow angle when you begin recording? 45° c x C a D A Recreation Geometry is used quite often in sailing. For example, to find the distance between the boat and a landmark on shore, sailors use a rule called doubling the angle on the bow. The rule says, measure the angle on the bow (the angle formed by your path and your line of sight to the landmark, also called your bearing) at point A. Check your bearing until, at point B, the bearing is double the reading at point A. The distance traveled from A to B is also the distance from the landmark to your new position. L A B LESSON 4.3 Triangle Inequalities 219 DG4TE_883_04.qxd 10/24/06 6:38 PM 21. By the Triangle Sum Conjecture, the third angle must measure 36° in the small triangle, but it measures 32° in the large triangle. These are the same angle, so they can’t have different measures. Page 220 Review In Exercises 19 and 20, calculate each lettered angle measure. 19. 21. What’s wrong with this picture of TRG? Explain. 20. d c f e 32 EXTENSIONS A. Ask students to use geometry software to explore congruence shortcuts. It’s especially useful for the one-step investigation. a d T g c b a h b a 90°, b 68°, 22 c 112°, d 112°, e 68°, f 56°, g 124°, h 124° 38 72 L 72 74 N 74 R G a 52°, b 38°, c 110°, d 35° In Exercises 22–24, complete the statement of congruence. B. Use Take Another Look activity 4 on page 255. ? 3.6 22. BAR ? FNK 4.2 23. FAR ABE HJ 3.6 24. HG cannot be ? determined HEJ N E A A J 52 B R G K 38 R F E H O RANDOM TRIANGLES Imagine you cut a 20 cm straw in two randomly selected places anywhere along You can use Fathom to generate many sets of random numbers quickly. You can also set up tables to view your data, and enter formulas to calculate quantities based on your data. its length. What is the probability that the three pieces will form a triangle? How do the locations of the cuts affect whether or not the pieces will form a triangle? Explore this situation by cutting a straw in different ways, or use geometry software to model different possibilities. Based on your informal exploration, predict the probability of the pieces forming a triangle. Now generate a large number of randomly chosen lengths to simulate the cutting of the straw. Analyze the results and calculate the probability based on your data. How close was your prediction? Your project should include Supporting the OUTCOMES After students have worked on the project, discuss how collecting more and more data (or pooling data) gets you closer and closer to the theoretical probability. Go to www.keymath.com/DG for a Fathom demonstration. (To avoid wasting straws, ask students to randomly bend pipe cleaners.) 220 Your prediction and an explanation of how you arrived at it. Your randomly generated data. An analysis of the results and your calculated probability. An explanation of how the location of the cuts affects the chances of a triangle being formed. Presentation of data is organized and clear. Explanations of predictions and descriptions of the results are consistent. If lengths are generated using a graphing calculator, Fathom, or another randomlength generator, the experimental probability for large samples will be around 25%. CHAPTER 4 Discovering and Proving Triangle Properties A graph of the sample space uses shading to show cut combinations that do produce a triangle. y 1 1/2 0 1/2 1 x