Section 2.2 Conditional Statements

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GoalsGoals

• Recognize and analyze a conditional statement• Write postulates about points, lines, and planes

using conditional statements

Conditional Statement

• A conditional statement has two parts,

• When conditional statements are written in if-then form, the part after the “if” is the __________, and the part after the “then” is the __________.

• Symbolic notation: p → q

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Examples

• If you are 13 years old, then you are a teenager.• Hypothesis:

• Conclusion:

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If there is snow on the ground, then it is cold.

Hypothesis:

Conclusion:

Rewrite in the if-then form An angle which measures 45 is acute.

A number divisible by 9 is also divisible by 3

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It is time for lunch if it is noon.

A triangle which is equilateral is also isosceles.

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True Statement

Statement which is always true

False Statement

Statement which is not always true

In order to prove something is false, we only need to show one example where it is false.

We call that example a ___________.

Counterexample – an example which proves a statement is false

Writing a Counterexample

• Write a counterexample to show that the following conditional statement is false– If x2 = 16, then x = 4.

– As a counterexample, let x =.• The hypothesis is _____, but the conclusion is

_____. • Therefore the conditional statement is _____.

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Converse• The converse of a conditional is formed by switching

the hypothesis and the conclusion.• The converse of p → q is q → p

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Write the converse of the following conditional statements:

Conditional: If I play football, then I am an athlete.

Converse:

Conditional: If two segments are congruent, then they have the same length.

Converse:

Negation

• The negative of the statement• Example: Write the negative of the statement– A is acute– A is ____ acute

• ~p represents “not p” or the negation of p

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Inverse Statements • Inverse– Negate the hypothesis and the conclusion of the

conditional statement– The inverse of p → q, is ~p → ~q

Write the inverse of the following Write the inverse of the following conditionals:conditionals:Conditional: If an angle measures 45Conditional: If an angle measures 45, then it is an acute , then it is an acute angle.angle.

Inverse:Inverse:

Conditional: If two segments are congruent, then they Conditional: If two segments are congruent, then they have the have the same length. same length.

Inverse:Inverse:

Contrapositive Statements

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ContrapositiveContrapositive Negate the hypothesis and the Negate the hypothesis and the

conclusion of the converseconclusion of the converse The contrapositive of The contrapositive of p p → q, is ~q → ~→ q, is ~q → ~p. p.

Write the contrapositive of the following Write the contrapositive of the following conditionals:conditionals:Conditional: If an angle measures 45Conditional: If an angle measures 45, then it is an acute , then it is an acute angle.angle.

Contrapositive:Contrapositive:

Conditional: If two segments are congruent, then they Conditional: If two segments are congruent, then they have the have the same length. same length.

Contrapositive:Contrapositive:

Example• Write the (a) converse, (b) inverse, and (c)

contrapositive of the statement.– If two angles are vertical, then the angles are

congruent.

• (a) Converse:

• (b) Inverse:

• (c) Contrapositive:11

Equivalent Statements

• When 2 statements are both true or both false• A conditional statement is equivalent to its

contrapositive.• The converse statement is equivalent to the inverse statement

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Conditional

Converse

Inverse

Contrapositive

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Definitions:All definitions can be read both forwards and backwards.

If two lines are perpendicular, then they intersect to form a right angle.

Forward:

Backward:

If two angles are complementary, then their sum is 90

Forward:

Backward:

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Conditional:

If two angles are supplementary, then the sum of the two angles is 180

Converse:

Conditional:

Converse:

If the sum of two angles is 180, then the two angles are supplementary.

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Biconditional StatementBiconditional Statement Is a statement that contains the phrase Is a statement that contains the phrase

“if and only if” “if and only if” This is equivalent to writing a This is equivalent to writing a

conditional statement and its conditional statement and its converseconverse

Can be either true or falseCan be either true or false To be true, both the conditional and To be true, both the conditional and

converse must be trueconverse must be true Symbolically: Symbolically: p ↔ q

Biconditional Statements

Biconditional

Two angles are supplementary if and only if their sum is 180.

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Writing a Postulate as a BiconditionalPostulate: (Conditional)

If P is in the interior of RST, then RSP + PST RST

Write the converse and decide if it is true.

Converse:

If RSP + PST RST, then P is in the interior of RST.

Combine it with the postulate to form a true biconditional.

Biconditional: