Postulate: A statement that is accepted without proof Theorem: An important statement that can be proven

2.2 Properties of Algebra proof – an argument which uses logic, definitions, properties, and previously proven statements When you write a proof, you must give a justification (reason) for each step to show that it is valid. For each justification, you can use a definition, postulate, property, or a piece of given information.

Algebraic Properties Foldable Make a hotdog fold. Take end and fold to center line. Open up all the folds and make a hamburger fold. Take other ends and fold it to center hamburger fold. Keeping paper like a hamburger, cut from the edge of the paper to the fold on each side. This should give you eight sections.

Addition Property of Equality Subtraction Property of Equality Multiplication Property of Equality Division Property of Equality Reflexive Property of Equality Symmetric Property of Equality Transitive Property of Equality Substitution Property of Equality

If a = b, then a + c = b + c (add. prop. =) Subtraction Property of Equality Multiplication Property of Equality Division Property of Equality Reflexive Property of Equality Symmetric Property of Equality Transitive Property of Equality Substitution Property of Equality

Addition Property of Equality If a = b, then a – c = b – c (subtr. prop. =) Multiplication Property of Equality Division Property of Equality Reflexive Property of Equality Symmetric Property of Equality Transitive Property of Equality Substitution Property of Equality

Addition Property of Equality Subtraction Property of Equality If a = b, then ac = bc (mult. prop. =) Division Property of Equality Reflexive Property of Equality Symmetric Property of Equality Transitive Property of Equality Substitution Property of Equality

Addition Property of Equality Subtraction Property of Equality Multiplication Property of Equality If a=b, then (div. prop. =) Reflexive Property of Equality Symmetric Property of Equality Transitive Property of Equality Substitution Property of Equality

Addition Property of Equality Subtraction Property of Equality Multiplication Property of Equality Division Property of Equality a = a (refl. prop. =) Symmetric Property of Equality Transitive Property of Equality Substitution Property of Equality

Addition Property of Equality Subtraction Property of Equality Multiplication Property of Equality Division Property of Equality Reflexive Property of Equality If a = b, then b = a (sym. prop. =) Transitive Property of Equality Substitution Property of Equality

Addition Property of Equality Subtraction Property of Equality Multiplication Property of Equality Division Property of Equality Reflexive Property of Equality Symmetric Property of Equality If a = b and b = c, then a = c (trans. prop. =) Substitution Property of Equality

Addition Property of Equality Subtraction Property of Equality Multiplication Property of Equality Division Property of Equality Reflexive Property of Equality Symmetric Property of Equality Transitive Property of Equality If a = b, then b can be substituted for a (subst. prop. =)

Example: Solve the equation 21 = 4x – = 4x – = 4x – = 4x x = 7 Solve algebraically Set up a proof

Example: Solve the equation 21 = 4x – 7. Write a justification for each step. Statements Reasons 1.) 21 = 4x – 7 Given 2.) 28 = 4x Add. prop. = 3.) 7 = x Div. prop. = Solve algebraically Set up a proof

Line segments with equal lengths are congruent, and angles with equal measures are also congruent. Therefore, the reflexive, symmetric, and transitive properties of equality have corresponding properties of congruence. Hotdog fold Open it up and hamburger fold Make shutters Cut one side of shutters into two sections.

Reflexive Property of Congruence Symmetric Property of Congruence Transitive Property of Congruence

fig. A  fig. A (refl. prop.  ) Symmetric Property of Congruence Transitive Property of Congruence

Reflexive Property of Congruence If fig. A  fig. B, then fig.B  fig.A (sym. prop.  ) Transitive Property of Congruence

Reflexive Property of Congruence Symmetric Property of Congruence If fig. A  fig. B and fig. B  fig. C, then figure A  figure C (trans. prop.  )

Example: Write a justification for each step. Given TA = ARDef.  segments 5y+6 = 2y+21Subst. prop. = 3y+6 = 21Subtr. prop. = 3y = 15Subtr. prop. = y = 5Div. prop. =  RAT 5y+62y+21