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Geometry Day 38

Geometry Day 38. Indirect Proof Inequalities in Two Triangles. Standards. Use methods of direct and indirect proof and determine whether a proof is logically valid.

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Geometry Day 38

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  1. Geometry Day 38 Indirect Proof Inequalities in Two Triangles

  2. Standards • Use methods of direct and indirect proof and determine whether a proof is logically valid. • Write geometric proofs, including proof by contradiction and proofs involving coordinate geometry. Use and compare a variety of ways to present deductive proofs, such as flow charts, paragraphs, two-column, and indirect proofs. • Apply the… Hinge Theorem

  3. Indirect Proof • Indirect reasoning is a form of logic that assumes a conclusion is false, then shows that this false conclusion leads to a contradiction. • Contrast with direct reasoning, which starts with given information and moves forward step-by-step until the conclusion is reached. • An indirect proof (also called a proof by contradiction) uses indirect reasoning to prove a mathematical concept: • Identify the conclusion you are being asked to prove. Make the assumption that this conclusion is false by assuming that the opposite is true. • Use logical reasoning to show that this assumption leads to a contradiction of the hypothesis, or some other fact such as a definition, postulate, theorem, or corollary. • Point out that since the assumption leads to a contradiction, the original conclusion (what you were asked to prove) must be true.

  4. Example • Use indirect reasoning to prove the Exterior Angle Inequality Theorem: • The exterior angle of a triangle is greater than either of the remote interior angles. (m3 > m1 and m3 > m2) • We assume that m3  m1 or m3  m2. 1 2 3

  5. Example • There are two possibilities: 1) m3 = m1 or 2) m3 < m1. • Case 1: m3 = m1 • m3 = m1 + m2 (Ext. Angle Th.) • m1 = m1 + m2 (Subst.) • 0 = m2 (Subtraction) 1 2 3

  6. Example • Case 2: m3 < m1 • m3 = m1 + m2 (Ext. Angle Th.) • The definition of inequality states that m3 must be greater than each of the angles added to make it, so m3 > m1 • In either case, there is acontradiction with theoriginal assumption,which means it must befalse. Therefore the original conclusion(m3 > m1 ) must betrue. 1 2 3

  7. More Indirect Proof • Indirect proofs can be used with algebra and number theory as well. • Prove: If –c is positive, then c is negative. • We assume that c is positive. • If c is positive then –c will be negative. However, this contradicts our given information. • Therefore the original conclusion must be correct: c is negative.

  8. More Indirect Proof • Prove: If the square of an integer is odd, then the integer is odd. • Note: Even integers can be written as 2n, and odd integers as 2n + 1. • We assume that the integer is even. • If we square an even integer: (2n)2 = (2n)(2n) = 4n2 • The result is divisible by 4, which means the square must be even. • This contradicts the given, so the integer must be odd.

  9. Inequalities in two triangles • Hinge Theorem: • If two sides of one triangle are congruent to two sides in another triangle, and the included angle in the first triangle is larger than the included angle in the second triangle, then the third side of the first triangle is longer than the third side of the second triangle. • Converse: • If two sides of one triangle are congruent to two sides in another triangle, and the third side of the first triangle is longer than the third side of the second triangle, then the included angle of the first triangle is larger than the included angle of the second triangle. • Examples (pp. 367-370)

  10. Homework 23 • Workbook: pp. 66, 70

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