04/20/23 Geometry 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.

CounterexampleCounterexample

Used to show a conditional statement is false.

It must keep the hypothesis true, It must keep the hypothesis true, but the conclusion false!but the conclusion false!

It must keep the hypothesis true, It must keep the hypothesis true, but the conclusion false!but the conclusion false!

It must keep the hypothesis true, It must keep the hypothesis true, but the conclusion false!but the conclusion false!

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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

ConverseConverse

Switch the hypothesis & conclusion parts of a conditional statement.

Ex: Write the converse of “If you are a brunette, then you have brown hair.”

If you have brown hair, then you are a brunette.

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Negation

The negative of the statement Writing the opposite of a statement. Example: Write the negative of the

statement A is acute A is not acute

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

InverseInverse

Negate the hypothesis & conclusion of a conditional statement.

Ex: Write the inverse of “If you are a brunette, then you have brown hair.”

If you are not a brunette, then you do not have brown hair.

ContrapositiveContrapositive

Negate, then switch the hypothesis & conclusion of a conditional statement.

Ex: Write the contrapositive of “If you are a brunette, then you have brown hair.”

If you do not have brown hair, then you are not a brunette.

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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