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Using Inductive Reasoning to Using Inductive Reasoning to 2-1 2-1 Make Conjectures Make Conjectures Section 2.1 Holt Geometry Holt McDougal Geometry Using Inductive Reasoning to 2-1 Make Conjectures Warm Up Complete each sentence. 1. 2. ? ? points are points that lie on the same line. points are points that lie in the same plane. 3. The sum of the measures of two Holt McDougal Geometry ? angles is 90°. Using Inductive Reasoning to 2-1 Make Conjectures • Inductive Reasoning is when one looks at several facts and then makes an educated guess on these facts. • That educated guess is called a conjecture. (Which may or may not be true.) Holt McDougal Geometry Using Inductive Reasoning to 2-1 Make Conjectures Find the next item in the pattern. January, March, May, ... Holt McDougal Geometry Using Inductive Reasoning to 2-1 Make Conjectures Find the next item in the pattern. 7, 14, 21, 28, … Holt McDougal Geometry Using Inductive Reasoning to 2-1 Make Conjectures Find the next item in the pattern. Holt McDougal Geometry Using Inductive Reasoning to 2-1 Make Conjectures Find the next item in the pattern 0.4, 0.04, 0.004, … Holt McDougal Geometry Using Inductive Reasoning to 2-1 Make Conjectures Complete the conjecture. The sum of two positive numbers is Holt McDougal Geometry ? . Using Inductive Reasoning to 2-1 Make Conjectures Complete the conjecture. The number of lines formed by 4 points, no three of which are collinear, is ? . Holt McDougal Geometry Using Inductive Reasoning to 2-1 Make Conjectures Complete the conjecture. The product of two odd numbers is Holt McDougal Geometry ? . Using Inductive Reasoning to 2-1 Make Conjectures To show that a conjecture is always true, you must prove it. To show that a conjecture is false, you have to find only one example in which the conjecture is not true. This case is called a counterexample. A counterexample can be a drawing, a statement, or a number. Holt McDougal Geometry Using Inductive Reasoning to 2-1 Make Conjectures Inductive Reasoning 1. Look for a pattern. 2. Make a conjecture. 3. Prove the conjecture or find a counterexample. Holt McDougal Geometry Using Inductive Reasoning to 2-1 Make Conjectures Show that the conjecture is false by finding a counterexample. For every integer n, n3 is positive. Holt McDougal Geometry Using Inductive Reasoning to 2-1 Make Conjectures Show that the conjecture is false by finding a counterexample. Two complementary angles are not congruent. Holt McDougal Geometry Using Inductive Reasoning to 2-1 Make Conjectures Show that the conjecture is false by finding a counterexample. The monthly high temperature in Abilene is never below 90°F for two months in a row. Monthly High Temperatures (ºF) in Abilene, Texas Jan Feb Mar Apr May Jun Jul Aug Sep 88 89 97 99 107 109 110 107 106 103 Holt McDougal Geometry Oct Nov Dec 92 89 Using Inductive Reasoning to 2-1 Make Conjectures Show that the conjecture is false by finding a counterexample. For any real number x, x2 ≥ x. Holt McDougal Geometry Using Inductive Reasoning to 2-1 Make Conjectures Show that the conjecture is false by finding a counterexample. Supplementary angles are adjacent. Holt McDougal Geometry Using Inductive Reasoning to 2-1 Make Conjectures Determine if each conjecture is true or false based on the information. If false, give a counter-example. 1.Given: AB, BC and CD Conjecture: A, B, C and D are collinear. 2. Given: ∠1 and ∠2 form a linear pair. Conjecture: m ∠1 + m ∠2 = 180 3.Given: ∠1 and ∠2 are supplementary. Conjecture: ∠1 = ∠2 Holt McDougal Geometry Using Inductive Reasoning to 2-1 Make Conjectures 4. Given: P, Q and R are collinear. Conjecture: Q is in between P and R. 5. Given: ∠1 and ∠2 are supplementary Given: ∠1 and ∠3 are supplementary Conjecture: ∠2 = ∠ 3 6. Given: points D, E, F, G Conjecture: D, E, F, G are noncollinear Holt McDougal Geometry Using Inductive Reasoning to 2-1 Make Conjectures 7. Given: Collinear points X, Y and Z Given: Z is between X and Y Conjecture: XY + YZ = XZ 8. Given: Noncollinear points L, M, N Conjecture: LM, MN and LN form a triangle 9. Given: PQRS is a rectangle Conjecture: PQ = RS and QR = SP Holt McDougal Geometry Using Inductive Reasoning to 2-1 Make Conjectures Lesson Quiz Find the next item in each pattern. 1. 0.7, 0.07, 0.007, … 2. Determine if each conjecture is true. If false, give a counterexample. 3. The quotient of two negative numbers is a positive number. 4. Every prime number is odd. 5. Two supplementary angles are not congruent. 6. The square of an odd integer is odd. Holt McDougal Geometry