Lesson 2.4-2.5

1. Lesson 2.4-2.5 Reasoning with Postulates and Algebra Properties

2. Postulates • Postulate 5- Through any two points there exists exactly one line • Postulate 6- A line contains at least two points • Postulate 7- If two lines intersect, then their intersection is exactly one point. • Postulate 8- Through any three noncollinear points there exists exactly one plane • Postulate 9- A plane contains at least three noncollinear points • Postualte 10- If two points lie in a plane, then the line containing them lies in the plane • Postulate 11- If two planes intersect, then their intersection is a line

3. Algebra Properties of Equality • Addition Property: If a = b, then a + c = b + c • Subtraction Property: If a = b, then a – c = b – c • Multiplication Property: If a = b, then ac = bc • Division Property: If a = b, then a/c = b/c as long as c does not equal zero • Substitution Property: If a = b, then a can be substituted in for b or vice versa in any equation for expression

4. Key Concepts • Reflexive Property of Equality • States the obvious to help begin a proof. For example if you want to show that AB = AB; you just say it because they are the same length. A 6 6 C D B 4 4 Since all corresponding sides and angles are congruent, and AB is congruent to AB because of the reflexive property of equality. Then Triangle ABC is congruent to Triangle ABD

5. Symmetric Property of Equality • Similar to a converse in that it “flips” • Ex: If AB = CD, then CD = AB by the symmetric property of equality • Transitive Property of Equality • Similar to “Law of Syllogism” • Ex: If AB = CD and CD = EF, then AB = EF by the transitive property of equality • Distributive Property a(b + c) = ab + ac -a(b + c) = -ab - ac a(b – c) = ab – ac -a(b – c) = -ab + ac