Geometry Honors Section 2.2

Introduction to Logic and Introduction to Deductive

Reasoning

The figures at the right are Venn diagrams. Venn diagrams are also called __________

diagrams, after the Swiss mathematician _______________________.

Which of the two diagrams correctly represents the statement

“If an animal is a whale, then it is a mammal”.

whale

whale

mammal

mammal

Euler

Leonard Euler

If-then statements like statement (1) are called *___________

In a conditional statement, the phrase following the word “if” is

the *_________. The phrase following the word “then” is the

*_________.

conditionals.

hypothesis

conclusion

If you interchange the hypothesis and the conclusion of a

conditional, you get the *converse of the original conditional.

Example 1: Write a conditional statement with the hypothesis “an animal is a reptile”

and the conclusion “the animal is a snake”. Is the statement true or false? If false, provide a

counterexample.

If an animal is a reptile than it is a snake.

crocodile :mplecounterexa

False

Write the converse of the conditional statement. Is the statement true or false?

If false, provide a counterexample.

If an animal is a snake, then it is a reptile.

True

Example 2: Consider the conditional statement “If two lines are perpendicular, then they

intersect to form a right angle”. Is the statement true or false? If false, provide a counterexample.

TRUE

Write the converse of the conditional statement. Is the statement true or false?

If false, provide a counterexample.

larperpendicu are lines

then theangle,right a form tointersect lines twoIf

TRUE

When an if-then statement and its converse are both true, we can combine the two statements into a single statement using the phrase “if and

only if” which is often abbreviated iff.

Example: Combine the two statements in example 2, into a single statement using iff.

anglesright form tointersect

they ifflar perpendicu are lines Two

Reasoning based on observing patterns, as we did in the first

section of Unit I, is called inductive reasoning. A serious drawback with

this type of reasoning is

your conclusion is not always true.

*Deductive reasoning is reasoning based on

Deductive reasoning

logically correct conclusions

always give a correct conclusion.

We will reason deductively by doing two column proofs. In the left hand column, we will have statements which lead from the given information to the conclusion which we are proving. In the right hand

column, we give a reason why each statement is true. Since we list the given

information first, our first reason will always be ______. Any other reason must be a _________, _________ or ________.

givendefinition postulate theorem

A theorem is a statement which can be proven.

We will prove our first theorems shortly.

Our first proofs will be algebraic proofs. Thus, we need to review

some algebraic properties. These properties, like postulates are

accepted as true without proof.

Reflexive Property of Equality:

Symmetric Property of Equality:

a = a

If a = b, then b = a

Addition Property of Equality:

Subtraction Property of Equality:

Multiplication Property of Equality:

If a = b, then a+c = b+c

If a = b, then a-c = b-c

If a = b, then ac = bc

Division Property of Equality:

cb c

a then 0,c and ba If

0?csay must weWhy

undefined! is 0by division Because

Substitution Property:

If two quantities are equal, then one may be substituted for the other in any equation or inequality.

Distributive Property (of Multiplication over Addition):

a(b+c) = ab + ac

Example: Complete this proof:

Given

Multiplication Property

Distributive Property

Addition Property

Division Property

Example: Prove the statement:

45x6-2x .1 1. Given

3

10 )

x