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Page 1 of 6 Triangles BEFORE Now WHY? You classified angles. You’ll classify triangles. So you can classify Research Triangle Park, as in Ex. 26. You can find the sum of the angle measures in a triangle. Word Watch acute triangle, p. 485 right triangle, p. 485 obtuse triangle, p. 485 congruent sides, p. 486 equilateral triangle, p. 486 isosceles triangle, p. 486 scalene triangle, p. 486 1 Cut a triangle from the corner of a piece of paper. Label the corners aA, aB, and aC. 2 Tear the corners off the triangle. 3 Rearrange aA, aB, and aC so that they are adjacent. Then make a conclusion about the angles of the triangle. In the activity, you found that the sum of the angle measures in TABC is 180. The sum of the angle measures in any triangle is 180. with EXAMPLE Reading Triangles are named by their vertices. The vertices of the triangle in the activity are A, B, and C, so the triangle can be named with the notation TABC. This notation is read “triangle ABC.” 1 Finding an Angle Measure in a Triangle Find the value of x in the triangle shown. Sum of angle measures in a triangle is 180. x 83 26 180 x 109 180 26 Add 83 and 26. x 71 Subtract 109 from each side. 83 x ANSWER The value of x is 71. Interior and Exterior Angles The three angles of any triangle are called interior angles. The sides of a triangle can be extended to form angles outside of the triangle that are adjacent and supplementary to the interior angles. These angles are called exterior angles. You can use the measures of interior angles to find the measures of exterior angles. exterior angle interior angles 484 Chapter 10 Geometric Figures Page 2 of 6 2 EXAMPLE Finding the Measure of an Exterior Angle Find the value of y in the figure. 55 To find the value of y, use the fact that adjacent interior and exterior angles of a triangle are supplementary. y 35 180 y 145 y 35 Definition of supplementary angles Subtract 35 from each side. ANSWER The value of y is 145. Your turn now 1. Find the value of x or y. 43 2. x 108 38 79 34 y You can classify any triangle by the measures of its interior angles. Classifying Triangles by Angle Measures Acute Triangle Right Triangle Obtuse Triangle An acute triangle has three acute angles. A right triangle has one right angle. An obtuse triangle has one obtuse angle. 60 50 120 35 70 EXAMPLE 3 50 25 40 Classifying a Triangle by Angle Measures Classify the triangle formed by the plant hanger by its angle measures. The triangle has one right angle, so it is a right triangle. Lesson 10.3 Triangles 485 Page 3 of 6 with Reading The notation AB & is read “line segment AB” and represents a line segment with endpoints A and B. The notation AB represents the length of AB &. Congruent Sides Just as congruent angles have the same measure, congruent sides of a triangle have the same length. You can use special marks to indicate that two sides or two angles of a triangle are congruent. In the triangle &* c XZ &* and at the right, the marks show that XY that aY c aZ. You can classify any triangle by the lengths of its sides. X Y Z Classifying Triangles by Side Lengths Equilateral Triangle Isosceles Triangle Scalene Triangle An equilateral triangle has 3 congruent sides. An isosceles triangle has at least 2 congruent sides. A scalene triangle has no congruent sides. 8m 9 in. 5 ft 9 in. 5m 4 EXAMPLE 11 ft 8m 9 in. 14 ft Classifying a Triangle by Side Lengths 36 in. Classify the triangle by the lengths of its sides. All three sides of the triangle are congruent, so the triangle is both isosceles and equilateral. Your turn now 36 in. Classify the triangle by its angle measures. 3. 4. 45 35 5. 80 53 100 60 36 in. 37 40 Classify the triangle by the lengths of its sides. 6. 7. 3 ft 2 ft 8. 18 cm 15 cm 6 cm 7m 7m 7m 2 ft 486 Chapter 10 Geometric Figures Page 4 of 6 INTERNET Exercises eWorkbook Plus CLASSZONE.COM More Practice, p. 714 Getting Ready to Practice Vocabulary In Exercises 1 and 2, tell whether the statement is true or false. Explain your reasoning. 1. A triangle that has one angle whose measure is greater than 90 is called an acute triangle. 2. An equilateral triangle can also be classified as an isosceles triangle. 3. The measures of two of the angles in a triangle are 90 and 57. Find the measure of the third angle. 4. Camera Tripod A tripod is used to keep a camera steady while taking pictures. A tripod has three legs that are each connected to a vertical shaft by a support bar. Find the value of y. 20 65 95 Classify the triangle by its angle measures. 5. 68, 22, 90 y 6. 53, 57, 70 Classify the triangle by the lengths of its sides. 7. 10 m, 12 m, 16 m 8. 12 in., 30 in., 30 in. Practice and Problem Solving with Example 1 2 3 4 Homework Exercises 9–11, 13–15 16–19 20–22 23–26 Online Resources Algebra Find the value of x. 9. 10. 11. 60 42 42 x 58 120 x 33 x 12. Find the Error Describe and correct the error in finding the value of x. CLASSZONE.COM • More Examples • eTutorial Plus x 56 180 x 56 x 124 The measures of two of the angles in a triangle are given. Find the measure of the third angle. 13. 112 and 34.8 14. 55.5 and 90.2 15. 128.7 and 30.4 Lesson 10.3 Triangles 487 Page 5 of 6 Algebra Find the value of y. 16. 17. 18. 92 y 44 44 y y 110 68 40 30 22 19. Drawbridge A castle has a drawbridge that can be raised or lowered. When the drawbridge is lowered, what angle does the chain form with the road leading up to the castle? Design 45 45 y Classify the triangle by its angle measures. 20. 68, 22, 90 21. 82, 64, 34 22. 135, 24, 21 Classify the triangle by the lengths of its sides. 23. 24. 12 cm ■ 6 cm 6 cm 6 cm 6 cm Drawbridge The Lasalle Street Bridge in Chicago is a drawbridge on the Chicago River. The drawbridge can be raised 70. When fully opened, what angle does the drawbridge form with the road leading up to the bridge? 25. 11 cm 6 cm 11 cm 11 cm 26. Research Triangle Park Duke University, the University of North Carolina, and North Carolina State University make up the three vertices of Research Triangle Park in North Carolina. The distances between the three universities are 9 miles, 21 miles, and 23 miles. Classify Research Triangle Park by the lengths of its sides. Algebra Find the values of x and y. 27. 28. 52 29. y x y 32 x 60 40 32 50 x y 30. Post and Beam In some houses, wall posts and ceiling beams are connected by support braces. Find the unknown angle measures in the post-and-beam design shown. 30 x y 31. Challenge The measure of one angle in a triangle is 36. The other two angles are congruent. What are their measures? Explain your reasoning. 488 Chapter 10 Geometric Figures Wall post e rac tb r o p Sup Ceiling beam Page 6 of 6 Mixed Review Order the numbers from least to greatest. (Lesson 4.7) 3 7 32. , 0.42, , 0.4 7 2 3 2 27 33. 6.64, 6 , 6 , 5 3 4 2 5 34. 3.25, 3 , 3.1, 3 3 7 35. Choose a Strategy Use a strategy from the list to solve the following problem. Explain your choice of strategy. You buy a prepaid phone card with a fixed number of minutes on it. You use one half of your minutes during the weekend and 6 minutes on Monday. If 24 minutes are left on your phone card, how many minutes did you start with? Problem Solving Strategies Draw a Diagram Write an Equation ■ Work Backward ■ ■ Basic Skills Use a straightedge and a compass to draw a segment whose length is the sum of the lengths of the two given segments. 36. 37. Test-Taking Practice INTERNET State Test Practice 38. Multiple Choice Which word describes a triangle whose angle measures are 42, 58, and 80? CLASSZONE.COM A. Right B. Obtuse C. Equilateral D. Acute 39. Short Response A triangle has two angles that measure 53 and 65. Find the measure of the third angle. Explain your reasoning. The Shape of Things Draw a square with a side length of 4 inches on a piece of paper. Divide the square into 16 smaller squares with side length 1 inch. Then mark off the lines shown in blue on the diagram. Cut carefully along these lines to produce the seven pieces of a tangram. Arrange all seven tangram pieces to form each figure. Lesson 10.3 Triangles 489