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Chapter 5 Relationships within Triangles In this chapter you will learn how special lines and segments in triangles relate.
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Chapter 5 Relationships within Triangles In this chapter you will learn how special lines and segments in triangles relate.

Jan 16, 2016

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Lionel Little
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Page 1: Chapter 5 Relationships within Triangles In this chapter you will learn how special lines and segments in triangles relate.

Chapter 5Relationships within Triangles

In this chapter you will learn how special lines and segments in

triangles relate.

Page 2: Chapter 5 Relationships within Triangles In this chapter you will learn how special lines and segments in triangles relate.

Objective: To use properties of midsegments to solve problems

Vocabulary: Midsegment of a triangle

Page 3: Chapter 5 Relationships within Triangles In this chapter you will learn how special lines and segments in triangles relate.
Page 4: Chapter 5 Relationships within Triangles In this chapter you will learn how special lines and segments in triangles relate.
Page 5: Chapter 5 Relationships within Triangles In this chapter you will learn how special lines and segments in triangles relate.
Page 6: Chapter 5 Relationships within Triangles In this chapter you will learn how special lines and segments in triangles relate.
Page 7: Chapter 5 Relationships within Triangles In this chapter you will learn how special lines and segments in triangles relate.
Page 8: Chapter 5 Relationships within Triangles In this chapter you will learn how special lines and segments in triangles relate.
Page 9: Chapter 5 Relationships within Triangles In this chapter you will learn how special lines and segments in triangles relate.
Page 10: Chapter 5 Relationships within Triangles In this chapter you will learn how special lines and segments in triangles relate.
Page 11: Chapter 5 Relationships within Triangles In this chapter you will learn how special lines and segments in triangles relate.
Page 12: Chapter 5 Relationships within Triangles In this chapter you will learn how special lines and segments in triangles relate.
Page 13: Chapter 5 Relationships within Triangles In this chapter you will learn how special lines and segments in triangles relate.
Page 14: Chapter 5 Relationships within Triangles In this chapter you will learn how special lines and segments in triangles relate.
Page 15: Chapter 5 Relationships within Triangles In this chapter you will learn how special lines and segments in triangles relate.
Page 16: Chapter 5 Relationships within Triangles In this chapter you will learn how special lines and segments in triangles relate.
Page 17: Chapter 5 Relationships within Triangles In this chapter you will learn how special lines and segments in triangles relate.
Page 18: Chapter 5 Relationships within Triangles In this chapter you will learn how special lines and segments in triangles relate.
Page 19: Chapter 5 Relationships within Triangles In this chapter you will learn how special lines and segments in triangles relate.
Page 20: Chapter 5 Relationships within Triangles In this chapter you will learn how special lines and segments in triangles relate.

Objective: To use properties of bisectors

Vocabulary:EquidistantDistance from a point to a line

Page 21: Chapter 5 Relationships within Triangles In this chapter you will learn how special lines and segments in triangles relate.
Page 22: Chapter 5 Relationships within Triangles In this chapter you will learn how special lines and segments in triangles relate.
Page 23: Chapter 5 Relationships within Triangles In this chapter you will learn how special lines and segments in triangles relate.
Page 24: Chapter 5 Relationships within Triangles In this chapter you will learn how special lines and segments in triangles relate.
Page 25: Chapter 5 Relationships within Triangles In this chapter you will learn how special lines and segments in triangles relate.
Page 26: Chapter 5 Relationships within Triangles In this chapter you will learn how special lines and segments in triangles relate.
Page 27: Chapter 5 Relationships within Triangles In this chapter you will learn how special lines and segments in triangles relate.
Page 28: Chapter 5 Relationships within Triangles In this chapter you will learn how special lines and segments in triangles relate.
Page 29: Chapter 5 Relationships within Triangles In this chapter you will learn how special lines and segments in triangles relate.
Page 30: Chapter 5 Relationships within Triangles In this chapter you will learn how special lines and segments in triangles relate.
Page 31: Chapter 5 Relationships within Triangles In this chapter you will learn how special lines and segments in triangles relate.
Page 32: Chapter 5 Relationships within Triangles In this chapter you will learn how special lines and segments in triangles relate.
Page 33: Chapter 5 Relationships within Triangles In this chapter you will learn how special lines and segments in triangles relate.
Page 34: Chapter 5 Relationships within Triangles In this chapter you will learn how special lines and segments in triangles relate.
Page 35: Chapter 5 Relationships within Triangles In this chapter you will learn how special lines and segments in triangles relate.
Page 36: Chapter 5 Relationships within Triangles In this chapter you will learn how special lines and segments in triangles relate.
Page 37: Chapter 5 Relationships within Triangles In this chapter you will learn how special lines and segments in triangles relate.
Page 38: Chapter 5 Relationships within Triangles In this chapter you will learn how special lines and segments in triangles relate.
Page 39: Chapter 5 Relationships within Triangles In this chapter you will learn how special lines and segments in triangles relate.
Page 40: Chapter 5 Relationships within Triangles In this chapter you will learn how special lines and segments in triangles relate.
Page 41: Chapter 5 Relationships within Triangles In this chapter you will learn how special lines and segments in triangles relate.

Objective: to identify properties of bisectors and angle bisectors

Vocabulary:•Concurrent•Point of concurrency•Circumcenter of a triangle•Circumscribed about•Incenter of a triangle•Inscribed in

Page 42: Chapter 5 Relationships within Triangles In this chapter you will learn how special lines and segments in triangles relate.
Page 43: Chapter 5 Relationships within Triangles In this chapter you will learn how special lines and segments in triangles relate.
Page 44: Chapter 5 Relationships within Triangles In this chapter you will learn how special lines and segments in triangles relate.
Page 45: Chapter 5 Relationships within Triangles In this chapter you will learn how special lines and segments in triangles relate.
Page 46: Chapter 5 Relationships within Triangles In this chapter you will learn how special lines and segments in triangles relate.
Page 47: Chapter 5 Relationships within Triangles In this chapter you will learn how special lines and segments in triangles relate.
Page 48: Chapter 5 Relationships within Triangles In this chapter you will learn how special lines and segments in triangles relate.
Page 49: Chapter 5 Relationships within Triangles In this chapter you will learn how special lines and segments in triangles relate.
Page 50: Chapter 5 Relationships within Triangles In this chapter you will learn how special lines and segments in triangles relate.
Page 51: Chapter 5 Relationships within Triangles In this chapter you will learn how special lines and segments in triangles relate.
Page 52: Chapter 5 Relationships within Triangles In this chapter you will learn how special lines and segments in triangles relate.
Page 53: Chapter 5 Relationships within Triangles In this chapter you will learn how special lines and segments in triangles relate.
Page 54: Chapter 5 Relationships within Triangles In this chapter you will learn how special lines and segments in triangles relate.
Page 55: Chapter 5 Relationships within Triangles In this chapter you will learn how special lines and segments in triangles relate.
Page 56: Chapter 5 Relationships within Triangles In this chapter you will learn how special lines and segments in triangles relate.
Page 57: Chapter 5 Relationships within Triangles In this chapter you will learn how special lines and segments in triangles relate.
Page 58: Chapter 5 Relationships within Triangles In this chapter you will learn how special lines and segments in triangles relate.
Page 59: Chapter 5 Relationships within Triangles In this chapter you will learn how special lines and segments in triangles relate.
Page 60: Chapter 5 Relationships within Triangles In this chapter you will learn how special lines and segments in triangles relate.
Page 61: Chapter 5 Relationships within Triangles In this chapter you will learn how special lines and segments in triangles relate.
Page 62: Chapter 5 Relationships within Triangles In this chapter you will learn how special lines and segments in triangles relate.
Page 63: Chapter 5 Relationships within Triangles In this chapter you will learn how special lines and segments in triangles relate.

Objective: To identify properties of medians and altitudes of a triangle

Vocabulary: Median of a triangleCentroid of a triangleAltitude of a triangleOrthocenter of a triangle

Page 64: Chapter 5 Relationships within Triangles In this chapter you will learn how special lines and segments in triangles relate.

In the Solve It!, the last set of segments drawn were the triangle’s medians. The median of a triangle is a segment Whose endpoints are the vertex and the midpoint of the Opposite side.

Page 65: Chapter 5 Relationships within Triangles In this chapter you will learn how special lines and segments in triangles relate.

In a triangle the point of concurrency of the medians is the centroid of the triangle. The point is also called the center of gravity of a triangle because it is a point where the triangular shape will balance. For any triangle the centroid is always inside the triangle.

Page 66: Chapter 5 Relationships within Triangles In this chapter you will learn how special lines and segments in triangles relate.
Page 67: Chapter 5 Relationships within Triangles In this chapter you will learn how special lines and segments in triangles relate.
Page 68: Chapter 5 Relationships within Triangles In this chapter you will learn how special lines and segments in triangles relate.

An altitude of a triangle is the perpendicular segment from a vertex of the triangle to the line containing the opposite side. An altitude of a triangle can be inside or outside the triangle, or it can be a side of the triangle.

Page 69: Chapter 5 Relationships within Triangles In this chapter you will learn how special lines and segments in triangles relate.
Page 70: Chapter 5 Relationships within Triangles In this chapter you will learn how special lines and segments in triangles relate.
Page 71: Chapter 5 Relationships within Triangles In this chapter you will learn how special lines and segments in triangles relate.

The lines that contain the altitudes of a triangle are concurrent at the orthocenter of the triangle. The orthocenter of a triangle can be inside, on, or outside the triangle.

Page 72: Chapter 5 Relationships within Triangles In this chapter you will learn how special lines and segments in triangles relate.
Page 73: Chapter 5 Relationships within Triangles In this chapter you will learn how special lines and segments in triangles relate.
Page 74: Chapter 5 Relationships within Triangles In this chapter you will learn how special lines and segments in triangles relate.
Page 75: Chapter 5 Relationships within Triangles In this chapter you will learn how special lines and segments in triangles relate.
Page 76: Chapter 5 Relationships within Triangles In this chapter you will learn how special lines and segments in triangles relate.
Page 77: Chapter 5 Relationships within Triangles In this chapter you will learn how special lines and segments in triangles relate.
Page 78: Chapter 5 Relationships within Triangles In this chapter you will learn how special lines and segments in triangles relate.
Page 79: Chapter 5 Relationships within Triangles In this chapter you will learn how special lines and segments in triangles relate.
Page 80: Chapter 5 Relationships within Triangles In this chapter you will learn how special lines and segments in triangles relate.
Page 81: Chapter 5 Relationships within Triangles In this chapter you will learn how special lines and segments in triangles relate.
Page 82: Chapter 5 Relationships within Triangles In this chapter you will learn how special lines and segments in triangles relate.
Page 83: Chapter 5 Relationships within Triangles In this chapter you will learn how special lines and segments in triangles relate.

Objective: To use indirect reasoning to write proofs

Vocabulary: Indirect reasoningIndirect proof

Page 84: Chapter 5 Relationships within Triangles In this chapter you will learn how special lines and segments in triangles relate.

In the Solve It!, you can conclude that a square must contain a certain number if you can eliminate the other three numbers as possibilities. In indirect reasoning, all possibilities are considered and then all but one are proved false.

Page 85: Chapter 5 Relationships within Triangles In this chapter you will learn how special lines and segments in triangles relate.

A proof involving indirect reasoning is an indirect proof. An indirect proof is sometimes called proof by contradiction.

Page 86: Chapter 5 Relationships within Triangles In this chapter you will learn how special lines and segments in triangles relate.

In the first step of an indirect proof you assume as true the opposite of what you want to prove.

Page 87: Chapter 5 Relationships within Triangles In this chapter you will learn how special lines and segments in triangles relate.
Page 88: Chapter 5 Relationships within Triangles In this chapter you will learn how special lines and segments in triangles relate.

To write an indirect proof, you have to be able to identify a contradiction.

Page 89: Chapter 5 Relationships within Triangles In this chapter you will learn how special lines and segments in triangles relate.
Page 90: Chapter 5 Relationships within Triangles In this chapter you will learn how special lines and segments in triangles relate.
Page 91: Chapter 5 Relationships within Triangles In this chapter you will learn how special lines and segments in triangles relate.
Page 92: Chapter 5 Relationships within Triangles In this chapter you will learn how special lines and segments in triangles relate.
Page 93: Chapter 5 Relationships within Triangles In this chapter you will learn how special lines and segments in triangles relate.
Page 94: Chapter 5 Relationships within Triangles In this chapter you will learn how special lines and segments in triangles relate.
Page 95: Chapter 5 Relationships within Triangles In this chapter you will learn how special lines and segments in triangles relate.
Page 96: Chapter 5 Relationships within Triangles In this chapter you will learn how special lines and segments in triangles relate.
Page 97: Chapter 5 Relationships within Triangles In this chapter you will learn how special lines and segments in triangles relate.
Page 98: Chapter 5 Relationships within Triangles In this chapter you will learn how special lines and segments in triangles relate.
Page 99: Chapter 5 Relationships within Triangles In this chapter you will learn how special lines and segments in triangles relate.

Objective: To use inequalities involving angles and sides of triangles

Vocabulary: No new vocabulary

Page 100: Chapter 5 Relationships within Triangles In this chapter you will learn how special lines and segments in triangles relate.

In the Solve It!, you explored triangles formed by various lengths of board. You may have noticed that changing the angle formed by the two sides of the sandbox changes the length of the third side.

Page 101: Chapter 5 Relationships within Triangles In this chapter you will learn how special lines and segments in triangles relate.
Page 102: Chapter 5 Relationships within Triangles In this chapter you will learn how special lines and segments in triangles relate.

The Comparison Property of Inequality allows you to prove the following corollary to the Triangle Exterior Angle Theorem:

Page 103: Chapter 5 Relationships within Triangles In this chapter you will learn how special lines and segments in triangles relate.
Page 104: Chapter 5 Relationships within Triangles In this chapter you will learn how special lines and segments in triangles relate.
Page 105: Chapter 5 Relationships within Triangles In this chapter you will learn how special lines and segments in triangles relate.
Page 106: Chapter 5 Relationships within Triangles In this chapter you will learn how special lines and segments in triangles relate.
Page 107: Chapter 5 Relationships within Triangles In this chapter you will learn how special lines and segments in triangles relate.
Page 108: Chapter 5 Relationships within Triangles In this chapter you will learn how special lines and segments in triangles relate.

For three segments to form a triangle, their lengths must be related in a certain way. Notice that only one of the sets of segments below can form a triangle. The sum of the smallest two lengths must be greater than the greatest length.

Page 109: Chapter 5 Relationships within Triangles In this chapter you will learn how special lines and segments in triangles relate.
Page 110: Chapter 5 Relationships within Triangles In this chapter you will learn how special lines and segments in triangles relate.
Page 111: Chapter 5 Relationships within Triangles In this chapter you will learn how special lines and segments in triangles relate.
Page 112: Chapter 5 Relationships within Triangles In this chapter you will learn how special lines and segments in triangles relate.
Page 113: Chapter 5 Relationships within Triangles In this chapter you will learn how special lines and segments in triangles relate.
Page 114: Chapter 5 Relationships within Triangles In this chapter you will learn how special lines and segments in triangles relate.
Page 115: Chapter 5 Relationships within Triangles In this chapter you will learn how special lines and segments in triangles relate.
Page 116: Chapter 5 Relationships within Triangles In this chapter you will learn how special lines and segments in triangles relate.
Page 117: Chapter 5 Relationships within Triangles In this chapter you will learn how special lines and segments in triangles relate.

Objective: To apply inequalities in two triangles

Vocabulary: No new vocabulary

Page 118: Chapter 5 Relationships within Triangles In this chapter you will learn how special lines and segments in triangles relate.

In the Solve It!, the hands of the clock and the segment labeled x form a triangle. As the time changes, the shape of the triangle changes, but the lengths of two of its sides do not change.

Page 119: Chapter 5 Relationships within Triangles In this chapter you will learn how special lines and segments in triangles relate.
Page 120: Chapter 5 Relationships within Triangles In this chapter you will learn how special lines and segments in triangles relate.
Page 121: Chapter 5 Relationships within Triangles In this chapter you will learn how special lines and segments in triangles relate.
Page 122: Chapter 5 Relationships within Triangles In this chapter you will learn how special lines and segments in triangles relate.
Page 123: Chapter 5 Relationships within Triangles In this chapter you will learn how special lines and segments in triangles relate.
Page 124: Chapter 5 Relationships within Triangles In this chapter you will learn how special lines and segments in triangles relate.
Page 125: Chapter 5 Relationships within Triangles In this chapter you will learn how special lines and segments in triangles relate.

The Converse of the Hinge Theorem is also true. The proof of the converse is an indirect proof.

Page 126: Chapter 5 Relationships within Triangles In this chapter you will learn how special lines and segments in triangles relate.
Page 127: Chapter 5 Relationships within Triangles In this chapter you will learn how special lines and segments in triangles relate.
Page 128: Chapter 5 Relationships within Triangles In this chapter you will learn how special lines and segments in triangles relate.
Page 129: Chapter 5 Relationships within Triangles In this chapter you will learn how special lines and segments in triangles relate.
Page 130: Chapter 5 Relationships within Triangles In this chapter you will learn how special lines and segments in triangles relate.
Page 131: Chapter 5 Relationships within Triangles In this chapter you will learn how special lines and segments in triangles relate.
Page 132: Chapter 5 Relationships within Triangles In this chapter you will learn how special lines and segments in triangles relate.
Page 133: Chapter 5 Relationships within Triangles In this chapter you will learn how special lines and segments in triangles relate.
Page 134: Chapter 5 Relationships within Triangles In this chapter you will learn how special lines and segments in triangles relate.
Page 135: Chapter 5 Relationships within Triangles In this chapter you will learn how special lines and segments in triangles relate.