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Unformatted text preview: 75 Inequalities in a Single Triangle 76 The Distance Between a Line and a Point. The Triangle Inequality
A) Theorem 75 a. If two sides of a triangle are not congruent, then the angles opposite them are not congruent and the larger angle is opposite the longer side b. Proof:
i. ii. Given: Prove: ABC AB AC C C B 1) 2) 3) 4) ABC AB AC 1) Given Let D be a point of mABD = mABC + mCBD ABD D AC , such that AD = AB 2) Point Plotting th. 3) 4) 5) 6) 7) 8) 5) mABC ABD 6) ABC D 7) D ACB 8) ABC ACB B) Theorem 76 a. If two angles of a triangle are not congruent, then the sides opposite them are not congruent and the longer side is opposite the larger angle i. If ii. Then 1. b. Example) i. Number 1 from the homework ii. 1. 2. GH KH KH GK 1) 2) 3) 4) Given: KH GK Prove: H is the smallest in the triangle 1) Given 2) 3) 4) GH KH C) Theorem 77 (The First Minimum Theorem) a. The shortest segment joining a point to a line is the perpendicular segment b. Proof: i. Given: PQ L at Q and R is any other point of L ii. Prove: PQ C PR 1) 2) 3) 4) 5) Q and R is any other point of L Q is a right angle R is acute
R Q PQ C PR PQ L at 1) Given 2) 3) 4) 5) D) Definition a. Distance i. Between a line and an external point is the length of the perpendicular segment from the point to the line. The distance between a line and a point on the line is defined to be zero E) Can any set of three segments form a triangle? F) Theorem 78 (The Triangle Inequality) a. The sum of the lengths of any two sides of a triangle is greater than the length of the third side b. Example) i. Can the following sets of numbers be the lengths of the sides of a triangle? 1. 3, 5, 8 2. 2, 2, 3 3. 8, 3, 9 ...
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 Spring '09
 Johnson
 Algebra, triangle, ABC AB AC, AC C C B

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