Objectives: 1) To develop an awareness of the structure of ... · 2-4 Reasoning in Algebra Objectives: 1) To develop an awareness of the structure of a mathematical system, connecting

2-4 Reasoning in Algebra

Objectives: 1) To develop an awareness of the structure of a mathematical system, connecting postulates, logical reasoning, and theorems.

2) To connect reasoning in algebra and geometry.




c c

c c

c c

c c

a

a

c

a

ab ac

Properties of Equality Both properties can be named APE, short for Addition Prop. of Equality.

Subtraction is actually adding the

negative of the number.

Both properties can be named MPE, short for Multiplication Prop. of

Equality. Division is actually

multiplying by the reciprocal of the

number.




Properties of Congruence

AB

A

AB

A

EF

C




There are three types of properties in mathematics:

and deduced. assumed, defining,

POSTULATES

A proof with a given and only one justified conclusion is often called a one-step proof.

Simplifying expressions and solving equations both represent a series of justified steps.

DEFINITIONS THEOREMS

Common Types

of Proofs:

two-column proof

paragraph proof

flow proof

Indirect proof

PROPERTIES

Corollaries

2-4 Reasoning in Algebra Objectives: 1) To develop an awareness of the structure of a mathematical system, connecting postulates, logical reasoning, and theorems.


Justify each step used to solve for 5x 12 = 32 + x .

APE (Addition Property of Equality)

APE

MPE (Multiplication Property of Equality)

Suppose that points A, B, and C are collinear with point B between points A and C.

Solve for x if AC = 21, AB =15 x , and BC = 4 + 2x . Justify each step.

A B C

15 x 4 + 2x Segment Addition Postulate

15 x 4 + 2x 21 Substitution Property

19 21

2 APE




Given

Angle

APE

Substitution Property

MPE

Recall that the length of segment AC is 21.

15 x 4 + 2x

A B C

AB = 13

Find the lengths of AB and BC by substituting x = 2 into the expressions in

The diagram. Check that AB + BC = 21.

BC = 8 AB + BC = 13 + 8 21 =




Name the property that justifies each statement.

The conclusion of the conditional statement shows the result after x is

from each side of the equation in the hypothesis. The property used is the ____________________.

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

The conclusion of the conditional statement is the same as the equation y + 4 = 3x (given) after x

has been substituted for y (given).The property used is the ____________________. Substitution Property

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

subtracted

APE




c. If P Q, Q R, and R S, then P S.

Transitive Property of Congruence

P R


P S.





Name the property the property of equality or congruence illustrated.

Reflexive Property of Congruence


Reasoning in Algebra 2-4

*Connect reasoning in algebra and geometry.

Angle Addition Postulate


Simplify

APE

MPE

Segment Addition Postulate


Distributive Property

Simplify

APE

MPE

textbook page 105


*Connect reasoning in algebra and geometry. textbook page 106

MPE


APE


APE

MPE

MPE

APE

Distributive Prop.

Symmetric Prop. of ≅

MPE Reflexive Prop. of Equality ( or = )

Symmetric Prop. of =

Reflexive Prop. of =

Substitution Prop. APE

Transitive Property of Congruence ( or ≅ )



15 5x

YU=AB K

PQR 3

EF + 7

WYT









End of Notes and Exercises for Lesson 2-4.

You may proceed to work on Practice 2-4 Worksheet, DNG page 281.

Please make an effort to print copies of the additional exercises for each lesson

as they will be good reference when you make your homework.

You use the library services to print these pages. I saved them all in PDF file and

You may choose to print 2-3 pages for each page of printing paper. Mr. Chapa,

The librarian or Ms. A. Rivera can help you on this matter.

Once again thank you for all your cooperation and work.

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Objectives: 1) To develop an awareness of the structure of ... · 2-4 Reasoning in Algebra Objectives: 1) To develop an awareness of the structure of a mathematical system, connecting