1. USING PROPERTIES FROM ALGEBRA ALGEBRAIC PROPERTIES OF EQUALITY Let a, b, and c be real numbers. SUBTRACTION PROPERTY ADDITION PROPERTY If a = b, then a + c = b + c. If a = b, then a - c = b - c. MULTIPLICATION PROPERTY DIVISION PROPERTY REFLECTIVE PROPERTY SYMMETRIC PROPERTY If a = b, then ac = bc. If a = b and c ≠ 0, then a  c = b  c. For any real number a, a = a. TRANSITIVE PROPERTY SUBSTITUTION PROPERTY If a = b, then b = a. If a = b and b = c, then a = c. If a = b, then a can be substituted for b in any equation or expression. 2.4

2. USING PROPERTIES FROM ALGEBRA PROPERTIES OF EQUALITY DISTANCE LENGTH REFLECTIVE SYMMETRIC For any distance AB, AB = AB. TRANSITIVE If AB = CD, then CD = AB. If AB = CD and CD = EF, then AB = EF. If m  A = m  B and, m  B = m  C, then m  A = m  C. 2.4 ANGLE MEASURE For any angle A, m  A = m  A. If m  A = m  B, then m  B = m  A.

3. USING PROPERTIES FROM GEOMETRY PROPERTIES OF CONGRUENCE AS THEOREMS SEGMENT CONGRUENCE REFLECTIVE SYMMETRIC  For any segment AB, AB  AB. TRANSITIVE  If AB  CD,  then CD  AB. If AB  CD and CD  EF, then AB  EF. If  A   B and,  B   C, then  A   C. 2.4 ANGLE CONGRUENCE For any angle A,   A   A. If  A   B, then  B   A.

4. COMBINING ALL ELEMENTS FROM GEOMETRY GEOMETRIC BOOK OF LAW LEGAL REASON FOR MAKING A STATEMENT. DEFINITIONS THEOREMS All definitions are Biconditional. POSTULATES Includes Formulas Or Axioms Inductive & Deductive Reasoning There are 2 laws of Deductive Reasoning: 1. Law of Detachment 2. Law of Syllogism 2.4 PROPERTIES LAWS OF LOGIC Of Equality & Congruence

5. Transitive Property of Angle Congruence Prove the Transitive Property of Congruence for angles. S OLUTION To prove the Transitive Property of Congruence for angles, begin by drawing three congruent angles. Label the vertices as A, B, and C. GIVEN A B, PROVE A CA C A B C B CB C

6. Transitive Property of Angle Congruence GIVEN A B, B CB C PROVE A CA C StatementsReasons 1 2 3 4 mA = mB 2. Definition of congruent angles 5 A  C 5. Definition of congruent angles A  B,1. Given B  C mB = mC 3. Definition of congruent angles mA = mC 4. Transitive property of equality

7. Using the Transitive Property This two-column proof uses the Transitive Property. StatementsReasons 2 3 4 m1 = m3 Definition of congruent angles GIVEN m3 = 40°,12,23 PROVE m1 = 40° 1 m1 = 40° Substitution property of equality 13 Transitive property of Congruence Givenm3 = 40°,12, 2323

8. Proving Theorem 2.3 THEOREM THEOREM 2.3 Right Angle Congruence Theorem All right angles are congruent. You can prove Theorem 2.3 as shown. GIVEN 1 and2 are right angles PROVE 1212

9. Proving Theorem 2.3 StatementsReasons 1 2 3 4 m1 = 90°, m2 = 90° Definition of right angles m1 = m2 Transitive property of equality 1  2 Definition of congruent angles GIVEN 1 and2 are right angles PROVE 1212 1 and2 are right angles Given