3. Find x. 4. FG = 6. m EFG = 7. EH = 8. m FGH = 9. m GFH = 10. m FEH =
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1 1/18 Warm Up Use the following diagram for numbers 1 2. The perpendicular bisectors of ABC meet at D. 1. Find DB. 2. Find AE. 22 B E A 14 D F G C B Use the following diagram for numbers 6. The angle bisectors of ABC meet at P. 3. Find x. 31 Use the following diagram and answer choices for #10-15 FH is the angle bisector of. E 10 cm F 32 H 5 cm EFG 4. FG = 6. m EFG = 7. EH = 8. m FGH = 9. m GFH = 10. m FEH = G January 20, 2016 Geometry 5.5 Inequalities in One Triangle 1
2 Geometry 6.5 Inequalities in One Triangle
3 6.5 & 6.6 Essential Question How are the sides related to the angles of a triangle? January 20, 2016 Geometry 5.1 Perpendiculars and Bisectors 3
4 Goals Use the Triangle Inequality Theorem. Be able to determine the largest and smallest angles and sides of a triangle. Use the Hinge Theorem and its converse to compare sides lengths and angle measures of two triangles. January 20, 2016 Geometry 5.5 Inequalities in One Triangle 4
5 January 20, 2016 Geometry 5.5 Inequalities in One Triangle 5
6 Can these three sides form a triangle? January 20, 2016 Geometry 5.5 Inequalities in One Triangle 6
7 Can these three sides form a triangle? No: leaves a gap of 1 in the middle. January 20, 2016 Geometry 5.5 Inequalities in One Triangle 7
8 January 20, 2016 Geometry 5.5 Inequalities in One Triangle 8
9 Can these three sides form a triangle? January 20, 2016 Geometry 5.5 Inequalities in One Triangle 9
10 2 2 5 January 20, 2016 Geometry 5.5 Inequalities in One Triangle 10
11 2 2 5 January 20, 2016 Geometry 5.5 Inequalities in One Triangle 11
12 2 2 5 January 20, 2016 Geometry 5.5 Inequalities in One Triangle 12
13 2 2 5 January 20, 2016 Geometry 5.5 Inequalities in One Triangle 13
14 2 2 5 January 20, 2016 Geometry 5.5 Inequalities in One Triangle 14
15 2 2 5 January 20, 2016 Geometry 5.5 Inequalities in One Triangle 15
16 2 2 5 January 20, 2016 Geometry 5.5 Inequalities in One Triangle 16
17 2 2 5 January 20, 2016 Geometry 5.5 Inequalities in One Triangle 17
18 2 2 5 January 20, 2016 Geometry 5.5 Inequalities in One Triangle 18
19 2 2 5 January 20, 2016 Geometry 5.5 Inequalities in One Triangle 19
20 No. Sides of 2 & 2 leave a gap of length 1. (Notice: < 5) January 20, 2016 Geometry 5.5 Inequalities in One Triangle 20
21 January 20, 2016 Geometry 5.5 Inequalities in One Triangle 21
22 Can these three sides form a triangle? January 20, 2016 Geometry 5.5 Inequalities in One Triangle 22
23 Can these three sides form a triangle? No: exactly matches the 5 leaving no room for an angle. January 20, 2016 Geometry 5.5 Inequalities in One Triangle 23
24 January 20, 2016 Geometry 5.5 Inequalities in One Triangle 24
25 Can these three sides form a triangle? January 20, 2016 Geometry 5.5 Inequalities in One Triangle 25
26 3 2 5 January 20, 2016 Geometry 5.5 Inequalities in One Triangle 26
27 3 2 5 January 20, 2016 Geometry 5.5 Inequalities in One Triangle 27
28 3 2 5 January 20, 2016 Geometry 5.5 Inequalities in One Triangle 28
29 3 2 5 January 20, 2016 Geometry 5.5 Inequalities in One Triangle 29
30 3 2 5 January 20, 2016 Geometry 5.5 Inequalities in One Triangle 30
31 3 2 5 January 20, 2016 Geometry 5.5 Inequalities in One Triangle 31
32 3 2 5 January 20, 2016 Geometry 5.5 Inequalities in One Triangle 32
33 3 2 5 January 20, 2016 Geometry 5.5 Inequalities in One Triangle 33
34 3 2 5 January 20, 2016 Geometry 5.5 Inequalities in One Triangle 34
35 3 2 5 January 20, 2016 Geometry 5.5 Inequalities in One Triangle 35
36 3 2 5 January 20, 2016 Geometry 5.5 Inequalities in One Triangle 36
37 3 2 5 January 20, 2016 Geometry 5.5 Inequalities in One Triangle 37
38 3 2 5 January 20, 2016 Geometry 5.5 Inequalities in One Triangle 38
39 3 2 5 January 20, 2016 Geometry 5.5 Inequalities in One Triangle 39
40 3 2 5 January 20, 2016 Geometry 5.5 Inequalities in One Triangle 40
41 3 2 5 January 20, 2016 Geometry 5.5 Inequalities in One Triangle 41
42 3 2 5 January 20, 2016 Geometry 5.5 Inequalities in One Triangle 42
43 No. Lengths of 3 & 2 fit the side of length 5 exactly. (Notice: = 5) January 20, 2016 Geometry 5.5 Inequalities in One Triangle 43
44 January 20, 2016 Geometry 5.5 Inequalities in One Triangle 44
45 Will these lengths form a triangle? January 20, 2016 Geometry 5.5 Inequalities in One Triangle 45
46 4 3 5 January 20, 2016 Geometry 5.5 Inequalities in One Triangle 46
47 4 3 5 January 20, 2016 Geometry 5.5 Inequalities in One Triangle 47
48 4 3 5 January 20, 2016 Geometry 5.5 Inequalities in One Triangle 48
49 4 3 5 January 20, 2016 Geometry 5.5 Inequalities in One Triangle 49
50 4 3 5 January 20, 2016 Geometry 5.5 Inequalities in One Triangle 50
51 4 3 5 January 20, 2016 Geometry 5.5 Inequalities in One Triangle 51
52 4 3 5 January 20, 2016 Geometry 5.5 Inequalities in One Triangle 52
53 4 3 5 January 20, 2016 Geometry 5.5 Inequalities in One Triangle 53
54 4 3 5 January 20, 2016 Geometry 5.5 Inequalities in One Triangle 54
55 Yes! January 20, 2016 Geometry 5.5 Inequalities in One Triangle 55
56 Together, lengths of 3 & 4 are longer than 5 and will meet before collapsing all the way. (Notice: > 5) January 20, 2016 Geometry 5.5 Inequalities in One Triangle 56
57 Triangle Inequality Theorem (6.11) In any triangle, the sum of any two sides is greater than the third side. a + b > c b + c > a a + c > b a b All of these must be true. c January 20, 2016 Geometry 5.5 Inequalities in One Triangle 57
58 Example 1 Is this triangle possible? Yes > > > January 20, 2016 Geometry 5.5 Inequalities in One Triangle 58
59 Example 2 Is this triangle possible? NO > > 4 But, > January 20, 2016 Geometry 5.5 Inequalities in One Triangle 59
60 Your Turn Is this triangle possible? Yes > > > January 20, 2016 Geometry 5.5 Inequalities in One Triangle 60
61 Given two sides, what is the range of the third side of a triangle? a = 3 b = 5 January 20, 2016 Geometry 5.5 Inequalities in One Triangle 61
62 Given two sides, what is the range of the third side of a triangle? Side c would be 2: just enough to cover side b. No Triangle. a = 3 b = 5 c = 2 January 20, 2016 Geometry 5.5 Inequalities in One Triangle 62
63 Rotate side a a = 3 b = 5 c = 2 January 20, 2016 Geometry 5.5 Inequalities in One Triangle 63
64 Rotate side a a = 3 b = 5 c = 2 January 20, 2016 Geometry 5.5 Inequalities in One Triangle 64
65 Rotate side a a = 3 c increases b = 5 January 20, 2016 Geometry 5.5 Inequalities in One Triangle 65
66 Rotate side a a = 3 c increases b = 5 January 20, 2016 Geometry 5.5 Inequalities in One Triangle 66
67 Rotate side a a = 3 c increases b = 5 January 20, 2016 Geometry 5.5 Inequalities in One Triangle 67
68 Rotate side a a = 3 c increases b = 5 January 20, 2016 Geometry 5.5 Inequalities in One Triangle 68
69 Rotate side a a = 3 c increases b = 5 January 20, 2016 Geometry 5.5 Inequalities in One Triangle 69
70 Rotate side a a = 3 c increases b = 5 January 20, 2016 Geometry 5.5 Inequalities in One Triangle 70
71 Rotate side a a = 3 c increases b = 5 January 20, 2016 Geometry 5.5 Inequalities in One Triangle 71
72 Rotate side a a = 3 c increases b = 5 January 20, 2016 Geometry 5.5 Inequalities in One Triangle 72
73 Rotate side a a = 3 c increases b = 5 January 20, 2016 Geometry 5.5 Inequalities in One Triangle 73
74 Rotate side a a = 3 c increases b = 5 January 20, 2016 Geometry 5.5 Inequalities in One Triangle 74
75 Rotate side a a = 3 c increases b = 5 January 20, 2016 Geometry 5.5 Inequalities in One Triangle 75
76 Rotate side a a = 3 c increases b = 5 January 20, 2016 Geometry 5.5 Inequalities in One Triangle 76
77 Rotate side a a = 3 c increases b = 5 January 20, 2016 Geometry 5.5 Inequalities in One Triangle 77
78 Rotate side a a = 3 c increases b = 5 January 20, 2016 Geometry 5.5 Inequalities in One Triangle 78
79 Rotate side a a = 3 c increases b = 5 January 20, 2016 Geometry 5.5 Inequalities in One Triangle 79
80 Rotate side a a = 3 c increases b = 5 January 20, 2016 Geometry 5.5 Inequalities in One Triangle 80
81 Rotate side a a = 3 c increases b = 5 January 20, 2016 Geometry 5.5 Inequalities in One Triangle 81
82 Rotate side a a = 3 c increases b = 5 January 20, 2016 Geometry 5.5 Inequalities in One Triangle 82
83 Rotate side a a = 3 c increases b = 5 January 20, 2016 Geometry 5.5 Inequalities in One Triangle 83
84 Rotate side a c increases a = 3 b = 5 January 20, 2016 Geometry 5.5 Inequalities in One Triangle 84
85 Rotate side a c increases a = 3 b = 5 January 20, 2016 Geometry 5.5 Inequalities in One Triangle 85
86 Rotate side a c increases a = 3 b = 5 January 20, 2016 Geometry 5.5 Inequalities in One Triangle 86
87 Rotate side a c increases a = 3 b = 5 January 20, 2016 Geometry 5.5 Inequalities in One Triangle 87
88 Rotate side a c increases a = 3 b = 5 January 20, 2016 Geometry 5.5 Inequalities in One Triangle 88
89 Rotate side a c =? a = 3 b = 5 January 20, 2016 Geometry 5.5 Inequalities in One Triangle 89
90 No triangle sides fall on top of each other. c = 8 a = 3 b = 5 January 20, 2016 Geometry 5.5 Inequalities in One Triangle 90
91 The extremes of c a = 3 b = 5 c = 2 c = 8 a = 3 b = 5 January 20, 2016 Geometry 5.5 Inequalities in One Triangle 91
92 Corollary to Triangle Inequality Theorem If two sides of a triangle measure a and b, with a being the larger side, then the third side, c, is greater than a b and less than a + b. a-b < c < a+b January 20, 2016 Geometry 5.5 Inequalities in One Triangle 92
93 Example 3 8 c 12 c is greater than 12 8 and less than January 20, 2016 Geometry 5.5 Inequalities in One Triangle 93
94 Example 3 8 c 12 c is greater than 4 and less than 20. Or, c is between 4 and 20. January 20, 2016 Geometry 5.5 Inequalities in One Triangle 94
95 Your Turn What are the possible values of x? x x is between 65 and 85. January 20, 2016 Geometry 5.5 Inequalities in One Triangle 95
96 Two Other Theorems 6.9: If one side of a triangle is longer than another side, then the angle opposite the larger side is larger than the angle opposite the shorter side. (Given the side lengths, the largest angle is opposite the longest side.) 6.10: If one angle of a triangle is larger than another angle, then the side opposite the larger angle is longer than the side opposite the smaller angle. (Given the angle measures, the longest side is opposite the largest angle.) January 20, 2016 Geometry 5.5 Inequalities in One Triangle 96
97 Example 4 List the angles of the triangle in order from smallest to largest. T R S S 14 8 R 15 T January 20, 2016 Geometry 5.5 Inequalities in One Triangle 97
98 Example 5 List the sides of the triangle in order from smallest to largest. b c a a 70 b 30 c 80 January 20, 2016 Geometry 5.5 Inequalities in One Triangle 98
99 The Hinge Theorem a c a d 1 2 b b Begin with two congruent triangles. (SAS) January 20, 2016 Geometry 5.6 Inequalities in Two Triangles 99
100 The Hinge Theorem a c a d 1 2 b b Rotate 1 to make it smaller. January 20, 2016 Geometry 5.6 Inequalities in Two Triangles 100
101 The Hinge Theorem a c a d 1 b 2 b Rotate 1 to make it smaller. January 20, 2016 Geometry 5.6 Inequalities in Two Triangles 101
102 The Hinge Theorem a c a d 1 b 2 b Rotate 1 to make it smaller. January 20, 2016 Geometry 5.6 Inequalities in Two Triangles 102
103 The Hinge Theorem a c a d 1 2 b b What happens to side c? January 20, 2016 Geometry 5.6 Inequalities in Two Triangles 103
104 The Hinge Theorem a c a d 1 2 b b What happens to side c? January 20, 2016 Geometry 5.6 Inequalities in Two Triangles 104
105 The Hinge Theorem a c a d 1 2 b b What happens to side c? January 20, 2016 Geometry 5.6 Inequalities in Two Triangles 105
106 The Hinge Theorem a c a d 1 2 b b What happens to side c? January 20, 2016 Geometry 5.6 Inequalities in Two Triangles 106
107 The Hinge Theorem a c a d 1 2 b b What happens to side c? January 20, 2016 Geometry 5.6 Inequalities in Two Triangles 107
108 The Hinge Theorem a c a d 1 2 b b What happens to side c? January 20, 2016 Geometry 5.6 Inequalities in Two Triangles 108
109 The Hinge Theorem a c a 1 2 b b d What happens to side c? January 20, 2016 Geometry 5.6 Inequalities in Two Triangles 109
110 The Hinge Theorem a c a 1 2 b b d What happens to side c? January 20, 2016 Geometry 5.6 Inequalities in Two Triangles 110
111 The Hinge Theorem a c a 1 2 b b d It gets smaller. January 20, 2016 Geometry 5.6 Inequalities in Two Triangles 111
112 The Hinge Theorem a c a 1 2 b b d Therefore, if 1 is smaller than 2, then side c is smaller than side d. January 20, 2016 Geometry 5.6 Inequalities in Two Triangles 112
113 Hinge Theorem If two sides of one triangle are congruent to two sides of another triangle, and the included angle of the first is larger than the included angle of the second, then the third side of the first is longer than the third side of the second. January 20, 2016 Geometry 5.6 Inequalities in Two Triangles 113
114 Converse of the Hinge Theorem If two sides of one triangle are congruent to two sides of another triangle, and the third side of the first is longer than the third side of the second, then the included angle of the first is larger than the included angle of the second. January 20, 2016 Geometry 5.6 Inequalities in Two Triangles 114
115 An Easy Memory Aid The smaller the angle, the smaller the opposite side. The larger the angle, the larger the opposite side. January 20, 2016 Geometry 5.6 Inequalities in Two Triangles 115
116 Think of a door. As the angle at the hinge increases, the size of the opening increases. January 20, 2016 Geometry 5.6 Inequalities in Two Triangles 116
117 Example 6 How does x compare to y? 5 x 5 y Since 30 < 90, x < y. January 20, 2016 Geometry 5.6 Inequalities in Two Triangles 117
118 Example 7 How does m A compare to m R? C T A 15 B R 15 S Since 16 < 17, m A < m R. January 20, 2016 Geometry 5.6 Inequalities in Two Triangles 118
119 Summary In a triangle, the sum of any two sides is greater than the third side. If two sides of a triangle measure a and b, then side c is between a b and a + b. In any triangle, the largest side is opposite the largest angle and the smallest side is opposite the smallest angle. Given two triangles with two congruent sides, the longer side is opposite the larger angle and the larger angle is opposite the longer side. January 20, 2016 Geometry 5.5 Inequalities in One Triangle 119
120 Homework 6.5 & 6.6 worksheet January 20, 2016 Geometry 5.5 Inequalities in One Triangle 120
121 The sum of any two sides of a triangle is greater than the remaining side. Will these three segments form a triangle? January 20, 2016 Geometry 5.5 Inequalities in One Triangle 121
Unit 4. Triangle Relationships. Oct 3 8:20 AM. Oct 3 8:21 AM. Oct 3 8:26 AM. Oct 3 8:28 AM. Oct 3 8:27 AM. Oct 3 8:27 AM
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