1. This chapter opens with a set of explorations designed to introduce you to new geometric topics that you will explore further. You will learn about the special properties right triangles as well as find their perimeters and areas. You will explore the relationships of the sides and angles of a triangle. You will also develop a deeper understanding of the Pythagorean theorem. The chapter ends with an exploration of logic and conditional statements.

5. 3.1 What is the Side Relationship With the Angle? Pg. 3 Triangle Inequality

6. 3.1 – What is the Side Relationship With the Angle? Triangle Inequality Today you are going to discover how the angles in a triangle relate to the sides of a triangle. You are also going to discover when three given side lengths don’t form a triangle.

7. 3.1 – TRIANGLE LENGTHS a. Examine at the triangle below. Measure the side lengths with a ruler and write it on the triangle. Then list the side lengths from smallest to largest. Then list the angles from smallest to largest.

8. 2cm 5cm 6.8cm 25° 10°145° Sides: _______________________ Angles: ________________________

9. b. What is the relationship between the largest side and the largest angle? What about the smallest side and the smallest angle? 2cm 5cm 6.8cm 25° 10°145°

10. b. What is the relationship between the largest side and the largest angle? What about the smallest side and the smallest angle? 2cm 5cm 6.8cm 25° 10°145° They are opposite from each other

11. c. Imagine that became smaller. Which side length would change? 2cm 5cm 6.8cm 25° 10°145°

12. d. Imagine that became larger. Which side length would change? 2cm 5cm 6.8cm 25° 10°145°

13. e. What is always the longest side in a right triangle? 2cm 5cm 6.8cm 25° 10°145°

14. Opposite Sides

15. 3.2 – ORDERING UP List the sides and angles in order from smallest to largest. Find the missing angles to help you.

16. Sides: ________________ Angles: _______________

17. Sides: _________________ Angles: ________________

18. 45°

19. 60°

20. 3.3 – CONSTRUCTING TRIANGLES Consider the segments below. Construct a triangle with the given side lengths.

22. 3.5 in

23. 3.4 – CONSTRUCTING TRIANGLES Consider the segments below. Construct a triangle with the given side lengths.

24. 3 in

25. 3.5 – CONSTRUCTING TRIANGLES Consider the segments below. Construct a triangle with the given side lengths.

26. 2.5 in

27. 3.6 – IS IT POSSIBLE? a. Use the manipulative provided by your teacher to investigate what is happening in the previous problem. Can a triangle be made with any three side lengths? If not, what condition(s) would make it impossible to build a triangle? Try building triangles with the side lengths provided by your teacher.

28. Trial #1: ______________ Trial #2: ______________ Trial #3: ______________ Make an equilateral triangle yes Make an isosceles triangle yes Make a scalene triangle yes

29. Trial #4: ______________ Trial #5: ______________ Trial #6: ______________ Make a triangle with sides of green, yellow, and blue (8.66cm, 10cm, 12.24cm) yes Make a triangle with sides of orange, purple, and red (5cm, 7.07cm, 14.14cm) no Make a triangle with sides of 2 orange and a yellow (5cm, 5cm, 10cm) no

30. b. For those triangles that could not be built, what happened? Why were they impossible? Two sides need to be long enough to reach

31. Triangle Inequality Theorem: a + b > c a + c > b b + c > a The sum of two sides needs to be greater than the third

32. http://hotmath.com/util/hm_flash_movie_fu ll.html?movie=/hotmath_help/gizmos/trian gleInequality.swf Triangle inequality

33. 3.7 – TRIANGLE IMPOSSIBLE Is it possible to construct a triangle with the given side lengths? If not, explain why not. a.3, 4, 5b.1, 4, 6 c. 17, 17, 33 d. 7, 45, 52 7 > 5 yes 5 > 6 no 34 > 33 yes 52 > 52 no

34. 3.8 – MAXIMUM AND MINIMUM LENGTHS Examine the pictures of the triangles below. There is a range of values that will complete a triangle. The fact that there are restrictions on the side of a triangle is referred to as the Triangle Inequality Theorem. Determine the minimum and maximum values that will make a triangle. What value does it have to be above? What value does it have to be below?

35. x + 13 > 19 x + 19 > 13 13 + 19 > x x > 6 x > -6 32 > x x < 32 More than 6 Less than 32 6 < x < 32

36. 15 – 14 <x< 15 + 14 1 <x< 29

37. 16 – 13 <x< 16 + 13 3 <x< 29

38. 3.9 – MAXIMUM AND MINIMUM LENGTHS Describe the possible lengths of the third side of the triangle given the lengths of the other two sides. a. 5m, 17m b.8in, 12in c. 10ft, 40ft

39. 17 – 5 <x< 17 + 5 12 <x< 22 a. 5m, 17m

40. 12 – 8 <x< 12 + 8 4 <x< 20 b.8in, 12in

41. 40 – 10 <x< 40 + 10 30 <x< 50 c. 10ft, 40ft

42. 3.10 – SMALLEST SIDE Use the information to determine what is the smallest whole number the following can be: 12 – 10 <x< 12 + 10 2 <x< 22

43. 25 – 15 <x< 25 + 15 10 <x< 40

44. 3.11 – POSSIBLE SIDE LENGTH Determine a possible length of the missing side of the triangle.

45. 7 – 5 <x< 7 + 5 2 <x< 12

46. 17 – 10 <x< 17 + 10 7 <x< 27

47. 3.12 – PERIMETER A student draws a triangle with a perimeter of 12in. The student says that the longest side measures 7in. How do you know that the student is incorrect? 7in P = 12in+5in 5 < 7