1. Some properties from algebra applied to geometry PropertySegmentsAngles Reflexive Symmetric Transitive PQ=QP m<1 = m<1 If AB= CD, then CD = AB. If m<A = m<B, then m<B = m<A If GH = JK and JK = LM, then GH = LM If m<1 = m<2 and m<2 = m<3, then m<1 = m<3

2. Examples Name the property of equality that justifies each statement. StatementReasons If AB + BC=DE + BC, then AB = DE m<ABC= m<ABC If XY = PQ and XY = RS, then PQ = RS If (1/3)x = 5, then x = 15 If 2x = 9, then x = 9/2 Subtraction property (=) Reflexive property (=) Substitution property (=) Multiplication property (=) Division property (=)

3. Example 2 Justify each step in solving 3x + 5 = 7 2 StatementReasons Given Multiplication property (=) Distributive property (=) Subtraction property (=) Division property (=) 3x + 5 = 7 2 2(3x + 5) = (7)2 2 3x + 5 = 14 3x = 9 x = 3 The previous example is a proof of the conditional: If 3x + 5 = 7, 2 then x=3 This type of proof is called a TWO-COLUMN PROOF

4. 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).

5. 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

6. 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 

7. 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