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Section 2.1 Conditional Statements

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Goals

Recognize and analyze a conditional statement

Write postulates about points, lines, and planes using conditional statements

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Conditional Statement

A conditional statement has two parts, a hypothesis and a conclusion.

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

p → q

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Examples

If you are 13 years old, then you are a teenager.

Hypothesis: You are 13 years old

Conclusion: You are a teenager

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Rewrite in the if-then form

All mammals breathe oxygen If an animal is a mammal, then it

breathes oxygen. A number divisible by 9 is also

divisible by 3 If a number s divisible by 9, then it

is divisible by 3.

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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 = -4.

The hypothesis is true, but the conclusion is false. Therefore the conditional statement is false.

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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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Negation

The negative of the statement Example: Write the negative of the

statement A is acute A is not acute

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

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Inverse and Contrapositive Inverse

Negate the hypothesis and the conclusion

The inverse of p → q, is ~p → ~q Contrapositive

Negate the hypothesis and the conclusion of the converse

The contrapositive of p → q, is ~q → ~p.

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

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

congruent. (a) Inverse: If 2 angles are not vertical, then

they are not congruent. (b) Converse: If 2 angles are congruent, then

they are vertical. (c) Contrapositive: If 2 angles are not

congruent, then they are not vertical.

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Equivalent Statements

When 2 statements are both true or both false

A conditional statement is equivalent to its contrapositive.

The inverse and the converse of any conditional are equivalent.

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Point, Line, and Plane Postulates Postulate 5: Through any two points there

exists exactly one line Postulate 6: A line contains at least two

points Postulate 7: If 2 lines intersect, then their

intersection is exactly one point Postulate 8: Through any three noncollinear

points there exists exactly one plane

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Postulate 9: A plane contains at least three noncollinear points

Postulate 10: If two points lie in a plane, then the line containing them lies in the plane

Postulate 11: If two planes intersect, then their intersection is a line