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
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.
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.
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
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
Transitive Property of Angle Congruence GIVEN A B, B CB C PROVE A CA C StatementsReasons 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
Using the Transitive Property This two-column proof uses the Transitive Property. StatementsReasons 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
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
Proving Theorem 2.3 StatementsReasons 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 and2 are right angles Given