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2.5 Reasoning with properties from Algebra GEOMETRY Goal 1: Using Properties from Algebra – Properties of Equality In all of the following properties – Let a, b, and c be real numbers 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 ca = cb Division property: If a = b, then a b for c 0 c c Addition Property This is the property that allows you to add the same number to both sides of an equation. STATEMENT x=5 REASON given 3+x=8 Addition property of equality Subtraction Property This is the property that allows you to subtract the same number to both sides of an equation. STATEMENT x=5 REASON given X-2=3 Subtraction property of equality Multiplication Property This is the property that allows you to multiply the same number to both sides of an equation. STATEMENT x=5 REASON given 3x = 15 Multiplication property of equality Division Property This is the property that allows you to divide the same number to both sides of an equation. STATEMENT x=5 x 5 3 3 REASON given Division property of equality More Properties of Equality Reflexive Property: a = a. Symmetric Property: If a = b, then b = a. Transitive Property: If a = b, and b = c, then a = c. Reflexive Property: a = a I know what you are thinking, duh this doesn’t seem too difficult to grasp. Just remember this one, when we begin to prove that triangles are congruent. STATEMENT x=x REASON Reflexive property of equality Symmetric Property: a = b so b = a I know another duh property. Just remember when you get an answer that is a little different than the one you are use to getting. (Do we like To always have x or y on the left side of the equal sign?) For example: 2 – y = 10 Transitive Property This one is many times confused with substitution property of equality. Remember transitive is like “transit” which means to move. Think of there being 3 bus stops: a, b, and c. If you move from a to b, then from b to c, it would have been the same as moving from a to c directly. STATEMENT REASON mA =43o given mB =43o given mA = mB Transitive property of equality Substitution Property of Equality If a = b, then a may be substituted for b in any equation or expression. You have used this many times in algebra. STATEMENT x=5 3+x=y 3+5=y REASON given given substitution property of equality Distributive Property a(b+c) = ab + ac ab + ac = a(b+c) STATEMENT mA + mA =90o 2mA =90o REASON given Distributive property Properties of Congruence Reflexive object A object A Symmetric If object A object B, then object B object A Transitive If object A object B and object B object C, then object A object C