SECTION 2.4: REASONING IN ALGEBRA

OBJECT IVE:• TO CONNECT REASONING IN ALGEBRA

AND GEOMETRY

REASONING IN ALGEBRA

• In Geometry, we accept postulates and properties as true.

• We use properties of equality to solve problems.

• We can justify each step of the problem solving using postulates and properties.

PROPERTIES OF EQUALITY

If a = b then a + c = b + c Addition Property of Equality

If a = b then a - c = b – c Subtraction Property of Equality

If a = b, then a ● c = b ● c Multiplication Property of Equality

If a = b, then , c ≠ 0 Division Property of Equality

a = a Reflexive Property of Equality

If a = b, then b = a Symmetric Property of Equality

If a = b and b = c, then a = c Transitive Property of Equality

c

b

c

a

MORE PROPERTIES OF EQUALITY

Substitution Property:

If a = b, then b can replace a in any expression

The Distributive Property:

a(b + c) = ab + bc

ACCEPTABLE JUSTIFICATIONS (WHY IS EACH STEP OF A PROBLEM TRUE??):

• Given Statements

• Postulates

• Properties of Equality or Congruence

• Definitions

EXAMPLE

Use the figure to solve for x. Justify each step.

Given: AC = 21

15-x 4+2x

AB + BC = AC

15-x + (4+2x) = 21

19+x= 21

x=2

EXAMPLE

Solve for x and justify each step.

Given m ABC = 128º

m ABD + m DBC = m ABC

x + 2x + 5 = 128

3x + 5 = 128

3x = 123

x = 41

PROPERTIES OF CONGRUENCE

Reflexive Property: AB AB A A

Symmetric Property: If AB CD, then CD AB

If A B, then B A

Transitive Property: If AB CD and CD EF, then AB EF

If A B and B C ,then A C

USING PROPERTIES OF EQUALITY AND CONGRUENCE

Name the property that justifies each statement.

a) If x = y and y + 4 = 3x, then x + 4 = 3x

b) If x + 4 = 3x, then 4 = 2x

c) If SPthenSRandRQQP ,,

Equality:

Compares 2 quantities

AB = CD and CD = EF, then

AB = EF

TRANSITIVE PROPERTY OF EQUALITY

(the lengths are equal)

Congruence:

Compares 2 geometric shapes

and

then

TRANSITIVE PROPERTY OF CONGRUENCE

(Segments are same size)

EQUALITY VS. CONGRUENCE

BmAm BA

CDAB EFCD EFAB