Unit 3Section 1

LogicalStatements

Conditional StatementsSuppose p and q are statements. Put them together in the form “____________”. We call this a __________________. When a conditional statement is written in this “if-then” form the “if” part is called the __________, which means “___________”. The “then” part is called the ____________, which means “________________________”.

Underline the antecedent with one line and the consequent with two lines, in each of these conditional statements.

If two lines intersect, then their intersection is exactly one point.

If two planes intersect then their intersection is a line.

If an animal is a tiger, then it has stripes.

If p then q

conditional statement

antecedent

to go beforeconseque

ntan event to follow another.

A prime number has exactly two divisors.

Seals swim.

All birds have feathers.

3n is odd if n is odd.

3n is odd only if n is odd.

Try writing these as if-then statements. In your new statement, underline the antecedent with one line and underline the consequent with two lines.

If a number is prime then it has exactly two divisors.

If an animal is a seal then it swims.

If an animal is a bird then it has feathers.If n is odd then 3n is odd.

If 3n is odd then n is odd.

When is a conditional statement false?For the next three statements, how would you show that the statement is false?

If you live in Kansas, then you live in Leavenworth.

If the product of two numbers is positive, then the two numbers must both be positive.

An example like this is called a _________________. The _____________ is true, but the ________________ is false in a counterexample.

Governor Brownback lives in Kansas, but not in Leavenworth.

(-2)(-8) = 16

counter exampleanteceden

tconsequent

Rewrite these false statements as if-then statements, and find or draw counterexamples to each one.

All musicians are guitar players.

The sum of two even numbers is odd.

Two rays that have the same endpoint are always opposite rays.

A number is prime only if it is odd.

If a person is a musician, then he/she is a guitar player. Piano

playerIf two numbers are even then their sum is odd. 2 + 4 = 6

EVEN

If two rays have the same endpoint then they are opposite rays.

If a number is prime then it is odd.

2 EVEN

The ConverseThe converse of a statement switches the __________ and the _____________

The converse of “If it is a tiger then it has stripes” is “____________________________”.

Is this true?____. Can you think of a counterexample? __________

antecedentconseque

nt

If it has stripes, then it is a tiger

NO!

Zebra

Statement Truth value/ Counterexample

Converse Truth Value/Counterexample

If a polygon is equilateral, the polygon is regular

An obtuse angle has a measure between 90° & 180

All ants are insects.

If a figure is a pentagon, then it is a decagon

F If a polygon is regular, then it is equilateral.

T

T If an angle is between 90 & 180, then it is an obtuse angle

T

For each statement below, decide its truth value. If it is false, write or draw a counter example. Then write the converse and decide its truth value

If an animal is an ant, then it is an insect.

TIf an animal is an insect, then it is an ant.

F

beetle

F If a figure is a decagon, then it is a pentagon

F

Is the truth value of the converse always the same as the truth value of the original statement? No!

Biconditional Statements

When both a statement and its converse are true, we can put them together into one statement called a bicondtional statement.

A biconditional statement uses the phrase “if and only if” between its two parts. Here is an example.

Two lines are perpendicular if and only if they intersect to form a right angle.

Note: the symbol for “perpendicular” in the language of geometry is

This means BOTH,• If two line are perpendicular, then they intersect to form a right

angle. AND the converse• If two lines intersect to form a right angle, then they are

perpendicular

Biconditional Statements cont.

In the converses that you wrote in the boxes on page 2, there was only one example where both the statement and the converse were true.

Notice that it was the definition of an obtuse angle.

A good definition can always be written as a biconditional statement.

If two angles are supplementary, then their measures sum to 180°

What biconditional statement can be made from this statement and its converse?

Converse:

Biconditonal:

If two angles have measures that sum to 180°, then they are supplementary

Two angles are supplementary if and only if their measures sum to 180°

Biconditional Statements cont.

What two statements can be made from these biconditional statements ?

If-Then

Converse

If BC and BA are opposite rays, then B is between A & C

If B is between A & C, then BC & BA are opposite rays

CABBABC and between is ifonly and if rays opposite are and

Biconditional Statements cont.

What two statements can be made from these biconditional statements ?

If- Then

Converse

If a polygon is a triangle, then it has three sides

If a polygon has three sides, then it is a triangle.

A polygon is a triangle if and only if it has three sides.