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PROOFS Geometry "measuring the earth“ is the branch of math that has to do with spatial relationships. A BIT OF HISTORY Geometry first became associated with land measurement in Egypt. The Egyptians were obliged to invent it in order to restore the landmarks that were destroyed by the periodic inundation of Nile River. Our textbook is based on Euclidean (or elementary) geometry. "Euclidean" (or "elementary") refers to a book written over 2,000 years ago called "The Elements" by a man named Euclid. Around 300 B.C. Euclid organized the Greek knowledge of geometry into a thirteen –volume work called ELEMENTS. His structure and method influence the way that geometry is taught today. Types of Reasoning 1. INDUCTIVE REASONNG - reaching a conclusion based on previous observation. - CONCLUSION IS PROBABLY TRUE BUT NOT NECESSARILY TRUE. EXAMPLES 1² = 1 1≤1 2² = 4 2≤4 CONSIDER THIS EXAMPLE (½) ²= ¼ ½> ¼ The conclusion does not hold true 2. Deductive Reasoning reaching a conclusion by combining known truths to create a new truth Deductive Reasoning deductive reasoning is certain, provided that the previously known truths are in fact true themselves. Types of Reasoning 1. INDUCTIVE REASONNG - reaching a conclusion based on previous observation. - CONCLUSION IS PROBABLY TRUE BUT NOT NECESSARILY TRUE. Types of Reasoning 1. DEDUCTIVE REASONNG - conclusion based on accepted statements (postulates, past theorems, given information, definition). Examples of Postulates/axioms Properties from algebra Linear pair - SUBSTITUTION postulate PROPERTY If two // lines are cut by a transversal, then corresponding angles are equal. - REFLEXIVE PROPERTY - SYMMETRIC PROPERTY - REFLEXIVE PROPERTY - SUBTRACTION PROPERTY - TRANSITIVE PROPERTY Examples of Definitions Midpoint of a segment Bisector of an angle Congruent angles and Segments Perpendicular lines Supplementary angles Complementary angles OTHERS Types of Reasoning 1. DEDUCTIVE REASONNG - conclusion must be true if hypothesis are true. KINDS OF PROOF DIRECT PROOF HAS 5 MAIN PARTS - THEOREM - GIVEN - PROVE STATEMENT - DIAGRAM - PROOF with STATEMENT & REASON INDIRECT PROOF USUALLY IN PARAGRAPH FORM THE OPPOSITE OF THE STATEMENT TO BE PROVEN IS ASSUMED TRUE, UNTIL THIS ASSUMPTION LEADS TO A CONTRADICTION. IT IS USED WHEN A DIRECT PROOF IS DIFFICULT OR IMPOSSIBLE. GIVEN: k // m: Transversal t cuts k and m. PROVE: 2 = 3 DIAGRAM/FIGURE t k 1 2 3 m STATEMENT REASON k // m: t is a transversal. Given Vertical angles are 1 = 2 equal If two parallel lines are 1 = 3 cut by a transversal, then corresponding angles are equal Transitive property 2 = 3