Lesson 2.4-2.5

Reasoning with Postulates and Algebra Properties

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

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

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

DB

C

6 6

4 4Since 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

• 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 - aca(b – c) = ab – ac -a(b – c) = -ab + ac