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Algebra Properties and Deductive Reasoning notes.notebook November 22, 2011 Algebra Properties (2.9) and Deductive Reasoning (2.10) Review of Algebra Properties Properties of Equality Addition Property: add the same amount to both sides Subtraction Property : subtract the same amount from both sides Multiplication Property : multiply by the same amount on both sides Division Property : divide both sides by the same amount Substitution Property: replace a value with an equal value 1 Algebra Properties and Deductive Reasoning notes.notebook November 22, 2011 Other useful properties Reflexive : a = a " something is equal to itself " Symmetric : If a = b , then b = a " state an equality either way " Transitive: If a = b and b = c, then a = c The properties of equality also relate to properties of congruence. For any segments AB, CD, and EF. For any angles R, S, and T. Reflexive Property Symmetric Property Transitive Property AB R If AB If R If AB If R AB R CD , then CD AB S , then S R CD, and CD EF, then AB EF. T. S, and S T, then R 2 Algebra Properties and Deductive Reasoning notes.notebook Theorem Proof November 22, 2011 statement accepted true with proof the use of deductive reasoning Deductive Reasoning You assume the hypothesis (given) is true and then write a series of statements with support (reasons) to lead to the conclusion (prove). Reasons given, definitions, postulates,and previously proven theorems and algebra properties Recall: Postulate statement accepted without proof Inductive reasoning using a pattern to make a conjecture Midpoint Theorem If point M is a midpoint of AB, then AM = AB hypothesis (given) Conclusion (prove) A statement reason 1. M is the midpoint of AB 2. AM = MB 3. AM + MB = AB 4. AM + AM = AB 2AM = AB 5. AM = AB M B 1. given 2. definition of midpoint 3. segment addition postulate 4. substitution property 5. division property 3 Algebra Properties and Deductive Reasoning notes.notebook November 22, 2011 Congruent Supplement Theorem Supplements of congruent angles are congruent. "If 2 angles are congruent, then their supplements are congruent." Given ≅ <B <A Statement Reason 1. <A <B A C 1. Given 2. m<A + m<C = 1800 m<B + m<D = 1800 2. def. of supp. angles 3. m<A + m<C = m<B + m<D 3. substitution 4. m<A + m<C = m<A + m<D 4. substitution 5. m<C = m<D 5. Subtraction property B D Right Angle Theorem All right angles are congruent. "If 2 angles are right angles then they are congruent." Given <A and <B are right angles. PROOF A B statement reason 2. m<A = 90 , m<B = 90 1. given 2. definition of right angle 3. <A <B 3. substitution property 1. <A and <B are right angles. 0 0 4 Algebra Properties and Deductive Reasoning notes.notebook November 22, 2011 HOMEWORK 2 WKS PROVE: Angle Bisector Theorem A (proof is very similar to the midpoint theorem proof from notes) If BX is the bisector of <ABC, then m<ABX = m<ABC X B C PROVE: Congruent Complement Theorem (proof is very similar to the congruent supplement theorem proof from notes) Complements of congruent angles are congruent. Given: A≅ B D A C B 5