• Lesson 22-2 ACTIVITY 22 Learning Targets:
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.