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Angles and Triangles ACTIVITY 22 Triangle Trivia Lesson 22-1 Properties of Triangles and Side Lengths My Notes Learning Targets: Determine when three side lengths form a triangle. Use the Triangle Inequality Property. Classify triangles by side length. • • • SUGGESTED LEARNING STRATEGIES: Interactive Word Wall, Summarizing, Look for a Pattern, Graphic Organizer Students in Mr. Mira’s math class made up some geometry games. Here are the rules for the game Matt and Allie created. Triangle Trivia Rules Properties of Triangles—Perimeter Variation Players: Materials: Directions: Three to four students Three number cubes and a “segment pieces” set of three each of the following lengths: 1 inch, 2 inches, 3 inches, 4 inches, 5 inches, and 6 inches. Take turns. Roll the three number cubes. Find a segment piece to match each number rolled. See whether a triangle can be formed from those segment pieces. The value of the perimeter of any triangle that can be formed is added to that player’s score. The first player to reach 50 points wins. © 2014 College Board. All rights reserved. Amir wonders what the game has to do with triangles. 1. Play the game above to see how it relates to triangles. Follow the rules. Record your results in the table. Player 1 Numbers Player 2 Score Numbers Player 3 Score Numbers Player 4 Score Numbers Score Activity 22 • Angles and Triangles 277 Lesson 22-1 Properties of Triangles and Side Lengths ACTIVITY 22 continued My Notes 2. There is more to the game than just adding numbers. How does the game relate to triangles? Amir noticed that he could tell whether the lengths would form a triangle even without the segment pieces. 3. Explain how Amir can determine whether a triangle can be formed from three given lengths. MATH TIP When sides of a figure have the same length, this can be shown by drawing marks, called tick marks, on those sides. For example, the equal sides of the isosceles and equilateral triangles in the table below right have the same number of tick marks. Matt and Allie’s game illustrates the following property that relates the side lengths of a triangle. Triangle Inequality Property For any triangle, the sum of any two sides must be greater than the length of the third side. Before students play another game, Mr. Mira wants to review the vocabulary terms scalene, isosceles, and equilateral with the class. He draws the following examples of triangles. Isosceles Triangles Equilateral Triangles © 2014 College Board. All rights reserved. Scalene Triangles 278 Unit 5 • Geometric Concepts Lesson 22-1 Properties of Triangles and Side Lengths Activity 22 continued My Notes 4. Based on Mr. Mira’s examples, describe each type of triangle. a. scalene triangle Math Tip b. isosceles triangle A triangle can be identified as scalene, isosceles, or equilateral by the lengths of its sides. c. equilateral triangle Amir creates a variation of Matt and Allie’s game. Here are the rules for Amir’s game. Triangle Trivia Rules - Name the Triangle Players: Materials: Directions: Three to four students Three number cubes Take turns rolling three number cubes. • If you can, form a scalene triangle .............add 5 points an isosceles triangle ........add 10 points an equilateral triangle .....add 15 points no triangle ............................add 0 points • If you make a mistake, deduct 10 points from your last correct score. • The first player to reach 25 points wins. © 2014 College Board. All rights reserved. 5. Make use of structure. When playing Amir’s variation of Triangle Trivia, suppose that the cubes landed on the following numbers. Tell how many points you would add to your score and why. a. 5, 5, 5 b. 1, 6, 4 c. 3, 2, 4 d.6, 6, 4 e. 1, 4, 1 Share your responses with your group members. Make notes as you listen to other members of your group. Ask and answer questions clearly to aid comprehension and to ensure understanding of all group members’ ideas. Activity 22 • Angles and Triangles 279 Lesson 22-1 Properties of Triangles and Side Lengths Activity 22 continued My Notes Check Your Understanding 6. Can a triangle be formed using the side lengths below? If so, is the triangle scalene, isosceles, or equilateral? Explain. a . 4 m, 4 m, and 8 m b. 8 ft, 6 ft, and 4 ft 7. If three segments form a triangle, what must be true about the sum of any two side lengths of the triangle? Lesson 22-1 Practice For Items 8–14, use the Triangle Inequality Property to determine whether a triangle can be formed with the given side lengths in inches. If a triangle can be formed, classify the triangle by the lengths of its sides. Explain your thinking. 8. a = 5, b = 5, c = 5 9. a = 3, b = 3, c = 7 10. a = 7, b = 4, c = 4 11. a = 8, b = 4, c = 5 12. a = 1, b = 2, c = 8 13. a = 8, b = 12, c = 4 15. Which of the following are possible side lengths of a triangle? A.12, 20, 15 B .33, 20, 12 C . 12, 20, 11 16. Reason abstractly. Is it necessary to find the sum of all three possible pairs of side lengths to use the Triangle Inequality Property when deciding if the sides form a triangle? Include an example in your explanation. 17. Construct viable arguments. Two sides of a triangle are 9 and 11 centimeters long. a. What is the shortest possible length for the third side? b. What is the longest possible length for the third side? 280 Unit 5 • Geometric Concepts © 2014 College Board. All rights reserved. 14. a = 12, b = 5, c = 13 Lesson 22-2 Properties of Triangles and Angle Measures ACTIVITY 22 continued Learning Targets: Classify angles by their measures. Classify triangles by their angles. Recognize the relationship between the lengths of sides and measures of angles in a triangle. Recognize the sum of angles in a triangle. • • • • My Notes MATH TIP If the rays are too short to measure with a protractor, extend the length of the sides of the angle. SUGGESTED LEARNING STRATEGIES: Interactive Word Wall, Summarizing, Visualization, Graphic Organizer Another way to classify triangles is by their angles. A right angle has a measure of 90°. An acute angle has a measure of less than 90°. An obtuse angle is greater than 90° and less than 180°. 1. Use the angles shown. C A B E © 2014 College Board. All rights reserved. D F a. Estimate the measure of each angle. ∠A ≈ ∠B ≈ ∠C ≈ ∠D ≈ ∠E ≈ ∠F ≈ b. Use appropriate tools strategically. Use a protractor to find the measure of each angle to the nearest degree. Then classify each angle as acute, obtuse, or right by its measure. ∠A = ∠B = ∠C = ∠D = ∠E = ∠F = Activity 22 • Angles and Triangles 281 Lesson 22-2 Properties of Triangles and Angle Measures Activity 22 continued My Notes Now Mr. Mira draws the following examples of triangles. Math Tip Acute Triangles A box at the vertex of an angle indicates an angle with measure 90°. A Right Triangles Obtuse Triangles E I D B G H C F 2. Based on Mr. Mira’s examples, describe each type of triangle. a. acute triangle b. obtuse triangle 3. A triangle can be labeled using both its angle measure and the lengths of its sides. a. Label the triangles that Mr. Mira drew by side length. b. Choose one of the triangles and give the two labels that describe it. c. Explain how the two labels together provide a better description of the triangle than either one alone. Share your ideas with our group and be sure to explain your thoughts using precise language and specific details to help group members understand your ideas and reasoning. 282 Unit 5 • Geometric Concepts © 2014 College Board. All rights reserved. c. right triangle Lesson 22-2 Properties of Triangles and Angle Measures Activity 22 continued Mr. Mira has his class investigate the sum of the measures of a triangle. Students measured the angles of some scalene, isosceles, and equilateral triangles. They recorded their results as shown. Isosceles Triangles Scalene Triangles 30° 135° 15° 20° 60° 45° 15° 70° 55° 90° 150° 20° 70° 85° 40° Equilateral Triangles 15° 40° 40° 120° My Notes 60° 90° 60° 45° 30° 75° 60° 60° 60° 60° 60° 70° 60° 75° 4. a. Find the sum of the angle measures for each triangle. The Triangle Sum Theorem states that the sum of the three angle measures in any triangle is always equal to a certain number. A theorem is a statement or conjecture that has been proven to be true. © 2014 College Board. All rights reserved. b. What is the sum of the angle measures in any triangle? MATH TERMS Activity 22 • Angles and Triangles 283 Lesson 22-2 Properties of Triangles and Angle Measures Activity 22 continued My Notes The Triangle Sum Theorem allows you to find the measure of the third angle in a triangle when you are given the other two angle measures. 5. Students played a game in which they chose two angle measures of a triangle and then determined the third angle measure. What must be true about the two angle measures the students choose? 6. Some of the angle measures students created for triangles are shown. For each pair of angle measures, find the measure of the third angle in the triangle. a . 43°, 94° b. 38°, 52° c. 57°, 39° d.140°, 12° © 2014 College Board. All rights reserved. e. 60°, 60° 284 Unit 5 • Geometric Concepts Lesson 22-2 Properties of Triangles and Angle Measures Activity 22 continued The angle measures of a triangle can be used to determine if the triangle is scalene, isosceles, or equilateral. Look back at the triangles Mr. Mira drew. Isosceles Triangles Scalene Triangles 30° 135° 15° 60° 45° 15° 70° 70° 85° 55° 90° 150° 20° 120° 40° Equilateral Triangles 15° 40° 40° 20° My Notes 60° 90° 60° 45° 30° 75° 60° 60° 60° 70° 60° 60° 60° 75° 7. Compare the angle measures of the triangles. Look for patterns in Mr. Mira’s examples to help you determine if the triangles described below are scalene, isosceles, or equilateral. a. a triangle with three different angle measures b. a triangle with exactly two congruent angle measures Math TERMS c. an equiangular triangle A triangle with three equal angles is called equiangular. © 2014 College Board. All rights reserved. 8. Look back at Item 6. Classify each triangle by its side length and by its angle measure. Another relationship exists between the angles and the sides of a triangle. In a triangle, the side opposite the angle with the greatest measure is the longest side. 9. Compare the angle measure to the side opposite the angle in a scalene triangle. What is true about the side opposite the angle with the least measure? Activity 22 • Angles and Triangles 285 Lesson 22-2 Properties of Triangles and Angle Measures ACTIVITY 22 continued My Notes Check Your Understanding For Items 10–12, sketch a triangle described by each pair of words below or state that it is not possible. Use tick marks and right angle symbols where appropriate. If it is not possible to sketch a triangle, explain why not. 10. scalene, obtuse 11. isosceles, acute 12. equilateral, right 13. Two angles in a triangle measure 35° and 50°. Explain how to find the measure of the third angle. LESSON 22-2 PRACTICE For Items 14–19, sketch a triangle described by each pair of words below or state that it is not possible. If it is not possible to sketch a triangle, explain why not. 14. scalene, right 15. isosceles, obtuse 16. equilateral, acute 17. isosceles, right 18. scalene, acute 19. equilateral, obtuse 21. Two angles in a triangle measure 65° each. What is the measure of the third angle? 22. Reason quantitatively and abstractly. Find the missing angle measure or measures in each triangle below. Then classify the triangle by both its angle measures and its side lengths. a. The three angles in a triangle have the same measure. b. Two angles in a triangle measure 45° each. c. Two angles in a triangle measure 25° and 50°. 23. Construct viable arguments. Determine whether each statement below is always true, sometimes true, or never true. Explain your reasoning. a. The acute angles of an isosceles triangle add up to 90°. b. An isosceles triangle has two equal angles. c. An equilateral triangle has a right angle. d. The largest angle of a scalene triangle can be opposite the shortest side. 286 Unit 5 • Geometric Concepts © 2014 College Board. All rights reserved. 20. Use appropriate tools strategically. Use a ruler and a protractor to sketch a triangle that is scalene and has an angle that measures 30°. Is the triangle acute, right, or obtuse? Explain.