2.5 Reasoning with properties from Algebra

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Transcript 2.5 Reasoning with properties from Algebra

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 REASON given

x

3  5 3 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 m  A =43 o m  B =43 o m  A = m  B given given 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 m  A + m  A =90 o 2m  A =90 o REASON given Distributive property

Properties of Congruence Reflexive object A  object A Symmetric If object A  Transitive object B, then object B  object A If object A  object B and object B  then object A  object C object C,