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Some properties from algebra applied to geometry Property Segments Angles Reflexive PQ=QP Symmetric If AB= CD, then CD = AB. If m<A = m<B, then m<B = m<A Transitive If GH = JK and JK = LM, then GH = LM If m<1 = m<2 and m<2 = m<3, then m<1 = m<3 m<1 = m<1 Examples Name the property of equality that justifies each statement. Statement Reasons If AB + BC=DE + BC, then AB = DE Subtraction property (=) m<ABC= m<ABC Reflexive property (=) If XY = PQ and XY = RS, then PQ = RS If (1/3)x = 5, then x = 15 If 2x = 9, then x = 9/2 Substitution property (=) Multiplication property (=) Division property (=) Example 2 Justify each step in solving 3x + 5 = 7 2 Statement The previous 3x + 5 = 7 example is a 2 proof of the 2(3x + 5) = (7)2 conditional: 2 If 3x + 5 = 7, 2 3x + 5 = 14 then x=3 Reasons Given Multiplication property (=) Distributive property (=) 3x = 9 Subtraction property (=) x=3 Division property (=) This type of proof is called a TWO-COLUMN PROOF Verifying Segment Relationships Five essential parts of a good proof: •State the theorem to be proved. •List the given information. •If possible, draw a diagram to illustrate the given information. •State what is to be proved. •Develop a system of deductive reasoning. (Use definitions, properties, postulates, undefined terms, or other theorems previously proved). Theorem 2.1 Congruence of segments is reflexive, symmetric, and transitive. Reflexive property: AB AB. Symmetric property: If AB CD, then CD AB Transitive property: If AB CD, and CD EF, then AB EF Abbreviation: reflexive prop. of segments symmetric prop. of segments transitive prop. of segments Verifying Angle Relationships Theorem 2-2 Supplement Theorem: If two angles form a linear pair, then they are supplementary angles Theorem 2-3: Congruence of angles is reflexive, symmetric and transitive. Abbreviation: reflexive prop. of <s symmetric prop. of <s transitive prop. of <s Theorem 2-4: Angles supplementary to the same angle or to congruent angles are congruent: Abbreviation: <s supp. to same < or <s are Verifying Angle Relationships Theorem 2-5: Angles complementary to the same angle or to congruent angles are congruent Abbreviation: <s comp. to same < or <s are Theorem 2-6 : All right angles are congruent Abbreviation: All rt. <s are Theorem 2-7: Vertical angles are congruent. Abbreviation: Vert. <s are Theorem 2-8: Perpendicular lines intersect to form four right angles. Abbreviation: lines form 4 rt. <s