1. GEOMETRY 2-1 Triangles Warm Up Classify each angle as acute, obtuse, or right. 1. 2. 3. 4. If the perimeter is 47, find x and the lengths of the three sides. right acute x = 5; 8; 16; 23 obtuse

2. GEOMETRY 2-1 Triangles In the figure, n is a whole number. What is the smallest possible value for n? A sewing club is making a quilt consisting of 25 squares with each side of the square measuring 30 centimeters. If the quilt has five rows and five columns, what is the perimeter of the quilt? 1. 2.

3. GEOMETRY 2-1 Triangles In the figure, n is a whole number. What is the smallest possible value for n? 1.

4. GEOMETRY 2-1 Triangles A sewing club is making a quilt consisting of 25 squares with each side of the square measuring 30 centimeters. If the quilt has five rows and five columns, what is the perimeter of the quilt? 2.

5. GEOMETRY 2-1 Triangles Classify triangles by their angle measures and side lengths. Use triangle classification to find angle measures and side lengths. Objectives

6. GEOMETRY 2-1 Triangles acute triangle equiangular triangle right triangle obtuse triangle equilateral triangle isosceles triangle scalene triangle Vocabulary

7. GEOMETRY 2-1 Triangles 12.0 Students find and use measures of sides and of interior and exterior angles of triangles and polygons to classify figures and solve problems. 15.0 Students use the Pythagorean theorem to determine distance and find missing lengths of sides of right triangles. California Standards

8. GEOMETRY 2-1 Triangles Recall that a triangle ( ) is a polygon with three sides. Triangles can be classified in two ways: by their angle measures or by their side lengths.

9. GEOMETRY 2-1 Triangles B A C AB, BC, and AC are the sides of ABC. A, B, C are the triangle's vertices.

10. GEOMETRY 2-1 Triangles Acute Triangle Three acute angles Triangle Classification By Angle Measures

11. GEOMETRY 2-1 Triangles Equilateral (Equiangular) Triangle Three congruent acute angles Triangle Classification By Angle Measures

12. GEOMETRY 2-1 Triangles Right Triangle One right angle Triangle Classification By Angle Measures

13. GEOMETRY 2-1 Triangles Obtuse Triangle One obtuse angle Triangle Classification By Angle Measures

14. GEOMETRY 2-1 Triangles Classify FHG by its angle measures. Teach! Example 1 EHG is a right angle. Therefore mEHF +mFHG = 90°. By substitution, 30°+ mFHG = 90°. So mFHG = 60°. FHG is an equilateral (Equiangular) triangle by definition.

15. GEOMETRY 2-1 Triangles Equilateral Triangle Three congruent sides Triangle Classification By Side Lengths

16. GEOMETRY 2-1 Triangles Isosceles Triangle At least two congruent sides Triangle Classification By Side Lengths

17. GEOMETRY 2-1 Triangles Scalene Triangle No congruent sides Triangle Classification By Side Lengths

18. GEOMETRY 2-1 Triangles Remember! When you look at a figure, you cannot assume segments are congruent based on appearance. They must be marked as congruent.

19. GEOMETRY 2-1 Triangles Classify EHF by its side lengths. Example 1: Classifying Triangles by Side Lengths From the figure,. So HF = 10, and EHF is isosceles.

20. GEOMETRY 2-1 Triangles Classify EHG by its side lengths. TEACH! : Classifying the Triangle by Side Lengths By the Segment Addition Postulate, EG = EF + FG = 10 + 4 = 14. Since no sides are congruent, EHG is scalene.

21. GEOMETRY 2-1 Triangles Classify ACD by its side lengths. TEACH! Example 3 From the figure,. So AC = 15, and ACD is scalene and probably obtuse.

22. GEOMETRY 2-1 Triangles Find the side lengths of JKL. Example 3: Using Triangle Classification Step 1 Find the value of x. Given. JK = KL Def. of  segs. 4x – 10.7 = 2x + 6.3 Substitute (4x – 10.7) for JK and (2x + 6.3) for KL. 2x = 17.0 x = 8.5 Add 10.7 and subtract 2x from both sides. Divide both sides by 2.

23. GEOMETRY 2-1 Triangles Find the side lengths of JKL. Example 3 Continued Step 2 Substitute 8.5 into the expressions to find the side lengths. JK = 4x – 10.7 = 4(8.5) – 10.7 = 23.3 KL = 2x + 6.3 = 2(8.5) + 6.3 = 23.3 JL = 5x + 2 = 5(8.5) + 2 = 44.5

24. GEOMETRY 2-1 Triangles Find the side lengths of equilateral FGH. TEACH! Example 3 Step 1 Find the value of y. Given. FG = GH = FH Def. of  segs. 3y – 4 = 2y + 3 Substitute (3y – 4) for FG and (2y + 3) for GH. y = 7 Add 4 and subtract 2y from both sides.

25. GEOMETRY 2-1 Triangles Find the side lengths of equilateral FGH. TEACH! Example 3 Continued Step 2 Substitute 7 into the expressions to find the side lengths. FG = 3y – 4 = 3(7) – 4 = 17 GH = 2y + 3 = 2(7) + 3 = 17 FH = 5y – 18 = 5(7) – 18 = 17

26. GEOMETRY 2-1 Triangles The amount of steel needed to make one triangle is equal to the perimeter P of the equilateral triangle. P = 3(18) P = 54 ft A steel mill produces roof supports by welding pieces of steel beams into equilateral triangles. Each side of the triangle is 18 feet long. How many triangles can be formed from 420 feet of steel beam? Example 4: Application

27. GEOMETRY 2-1 Triangles A steel mill produces roof supports by welding pieces of steel beams into equilateral triangles. Each side of the triangle is 18 feet long. How many triangles can be formed from 420 feet of steel beam? Example 4: Application Continued To find the number of triangles that can be made from 420 feet of steel beam, divide 420 by the amount of steel needed for one triangle. 420  54 = 7 triangles 7979 There is not enough steel to complete an eighth triangle. So the steel mill can make 7 triangles from a 420 ft. piece of steel beam.

28. GEOMETRY 2-1 Triangles The amount of steel needed to make one triangle is equal to the perimeter P of the equilateral triangle. P = 3(7) P = 21 in. Each measure is the side length of an equilateral triangle. Determine how many 7 in. triangles can be formed from a 100 in. piece of steel. TEACH! Example 4

29. GEOMETRY 2-1 Triangles To find the number of triangles that can be made from 100 inches of steel, divide 100 by the amount of steel needed for one triangle. 100  7 = 14 triangles 2727 There is not enough steel to complete a fifteenth triangle. So the manufacturer can make 14 triangles from a 100 in. piece of steel. Each measure is the side length of an equilateral triangle. Determine how many 7 in. triangles can be formed from a 100 in. piece of steel. TEACH! Example 4 Continued

30. GEOMETRY 2-1 Triangles The amount of steel needed to make one triangle is equal to the perimeter P of the equilateral triangle. P = 3(10) P = 30 in. Each measure is the side length of an equilateral triangle. Determine how many 10 in. triangles can be formed from a 100 in. piece of steel. TEACH! Example 5

31. GEOMETRY 2-1 Triangles To find the number of triangles that can be made from 100 inches of steel, divide 100 by the amount of steel needed for one triangle. 100  10 = 10 triangles The manufacturer can make 10 triangles from a 100 in. piece of steel. Each measure is the side length of an equilateral triangle. Determine how many 10 in. triangles can be formed from a 100 in. piece of steel. TEACH! Example 5 Continued

32. GEOMETRY 2-1 Triangles Lesson Quiz Classify each triangle by its angles and sides. 1. MNQ 2. NQP 3. MNP 4. Find the side lengths of the triangle. acute; equilateral obtuse; scalene acute; scalene 29; 29; 23

33. GEOMETRY 2-1 Triangles

34. GEOMETRY 2-1 Triangles 150° 73° 1; Parallel Post. Warm Up 1. Find the measure of exterior DBA of BCD, if mDBC = 30°, mC= 70°, and mD = 80°. 2. What is the complement of an angle with measure 17°? 3. How many lines can be drawn through N parallel to MP? Why?

35. GEOMETRY 2-1 Triangles Find the measures of interior and exterior angles of triangles. Apply theorems about the interior and exterior angles of triangles. Objectives

36. GEOMETRY 2-1 Triangles auxiliary line corollary interior exterior interior angle exterior angle remote interior angle Vocabulary

37. GEOMETRY 2-1 Triangles

38. GEOMETRY 2-1 Triangles An auxiliary line is a line that is added to a figure to aid in a proof. An auxiliary line used in the Triangle Sum Theorem

39. GEOMETRY 2-1 Triangles After an accident, the positions of cars are measured by law enforcement to investigate the collision. Use the diagram drawn from the information collected to find mXYZ. Example 6A: Application mXYZ + mYZX + mZXY = 180° Sum. Thm mXYZ + 40 + 62 = 180 Substitute 40 for mYZX and 62 for mZXY. mXYZ + 102 = 180 Simplify. mXYZ = 78° Subtract 102 from both sides.

40. GEOMETRY 2-1 Triangles After an accident, the positions of cars are measured by law enforcement to investigate the collision. Use the diagram drawn from the information collected to find mYWZ. Example 6B: Application mYXZ + mWXY = 180° Lin. Pair Thm. and  Add. Post. 62 + mWXY = 180 Substitute 62 for mYXZ. mWXY = 118° Subtract 62 from both sides. Step 1 Find m  WXY. 118°

41. GEOMETRY 2-1 Triangles After an accident, the positions of cars are measured by law enforcement to investigate the collision. Use the diagram drawn from the information collected to find mYWZ. Example 6B: Application Continued Step 2 Find mYWZ. 118° mYWX + mWXY + mXYW = 180° Sum. Thm mYWX + 118 + 12 = 180 Substitute 118 for mWXY and 12 for mXYW. mYWX + 130 = 180 Simplify. mYWX = 50° Subtract 130 from both sides.

42. GEOMETRY 2-1 Triangles Use the diagram to find mMJK. TEACH! Example 6 mMJK + mJKM + mKMJ = 180° Sum. Thm mMJK + 104 + 44= 180 Substitute 104 for mJKM and 44 for mKMJ. mMJK + 148 = 180 Simplify. mMJK = 32° Subtract 148 from both sides.

43. GEOMETRY 2-1 Triangles A corollary is a theorem whose proof follows directly from another theorem. Here are two corollaries to the Triangle Sum Theorem.

44. GEOMETRY 2-1 Triangles One of the acute angles in a right triangle measures 2x°. What is the measure of the other acute angle? Example 7: Finding Angle Measures in Right Triangles mA + mB = 90° 2x + mB = 90 Substitute 2x for mA. mB = (90 – 2x)° Subtract 2x from both sides. Let the acute angles be  A and  B, with m  A = 2x°. Acute s of rt. are comp.

45. GEOMETRY 2-1 Triangles The measure of one of the acute angles in a right triangle is 63.7°. What is the measure of the other acute angle? TEACH! Example 7a mA + mB = 90° 63.7 + mB = 90 Substitute 63.7 for mA. mB = 26.3° Subtract 63.7 from both sides. Let the acute angles be  A and  B, with m  A = 63.7°. Acute s of rt. are comp.

46. GEOMETRY 2-1 Triangles The measure of one of the acute angles in a right triangle is x°. What is the measure of the other acute angle? TEACH! Example 7b mA + mB = 90° x + mB = 90 Substitute x for mA. mB = (90 – x)° Subtract x from both sides. Let the acute angles be  A and  B, with m  A = x°. Acute s of rt. are comp.

47. GEOMETRY 2-1 Triangles The measure of one of the acute angles in a right triangle is 48. What is the measure of the other acute angle? TEACH! Example 7c mA + mB = 90° Acute s of rt. are comp. 2° 5 Let the acute angles be A and B, with mA = 48. 2° 5 Subtract 48 from both sides. 2525 Substitute 48 for mA. 2525 48 + mB = 90 2525 mB = 41 3° 5

48. GEOMETRY 2-1 Triangles The interior is the set of all points inside the figure. The exterior is the set of all points outside the figure. Interior Exterior

49. GEOMETRY 2-1 Triangles An interior angle is formed by two sides of a triangle. An exterior angle is formed by one side of the triangle and extension of an adjacent side. Interior Exterior 4 is an exterior angle. 3 is an interior angle.

50. GEOMETRY 2-1 Triangles Each exterior angle has two remote interior angles. A remote interior angle is an interior angle that is not adjacent to the exterior angle. Interior Exterior 3 is an interior angle. 4 is an exterior angle. The remote interior angles of 4 are 1 and 2.

51. GEOMETRY 2-1 Triangles Corollary: The measure of an exterior angle of a triangle is greater than the measure of either of its remote angles.

52. GEOMETRY 2-1 Triangles Find mB. Example 8: Applying the Exterior Angle Theorem mA + mB = mBCD Ext.  Thm. 15 + 2x + 3 = 5x – 60 Substitute 15 for mA, 2x + 3 for mB, and 5x – 60 for mBCD. 2x + 18 = 5x – 60 Simplify. 78 = 3x Subtract 2x and add 60 to both sides. 26 = x Divide by 3. mB = 2x + 3 = 2(26) + 3 = 55°

53. GEOMETRY 2-1 Triangles Find mACD. TEACH! Example 8 mACD = mA + mB Ext.  Thm. 6z – 9 = 2z + 1 + 90 Substitute 6z – 9 for mACD, 2z + 1 for mA, and 90 for mB. 6z – 9 = 2z + 91 Simplify. 4z = 100 Subtract 2z and add 9 to both sides. z = 25 Divide by 4. mACD = 6z – 9 = 6(25) – 9 = 141°

54. GEOMETRY 2-1 Triangles

55. GEOMETRY 2-1 Triangles Find mK and mJ. Example 9: Applying the Third Angles Theorem K  JK  J mK = mJ 4y 2 = 6y 2 – 40 –2y 2 = –40 y 2 = 20 So mK = 4y 2 = 4(20) = 80°. Since mJ = mK, mJ = 80°. Third s Thm. Def. of  s. Substitute 4y 2 for mK and 6y 2 – 40 for mJ. Subtract 6y 2 from both sides. Divide both sides by -2.

56. GEOMETRY 2-1 Triangles TEACH! Example 9 Find mP and mT. P  TP  T mP = mT 2x 2 = 4x 2 – 32 –2x 2 = –32 x 2 = 16 So mP = 2x 2 = 2(16) = 32°. Since mP = mT, mT = 32°. Third s Thm. Def. of  s. Substitute 2x 2 for mP and 4x 2 – 32 for mT. Subtract 4x 2 from both sides. Divide both sides by -2.

57. GEOMETRY 2-1 Triangles Lesson Quiz: Part I 1. The measure of one of the acute angles in a right triangle is 56 °. What is the measure of the other acute angle? 2. Find mABD. 3. Find mN and mP. 124° 75°; 75° 2323 33 ° 1313

58. GEOMETRY 2-1 Triangles Lesson Quiz: Part II 4. The diagram is a map showing John's house, Kay's house, and the grocery store. What is the angle the two houses make with the store? 30°