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IM3 Mr. Smith

- Inverse -  Converse -  Contrapositive

Conditional Statements

+ Conditional Statement

Definition: A conditional statement is a statement that can be written in if-then form. “If _________, then __________.”

Example 1: If mom went to the store after school, then mom will buy something.

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Conditional Statements have two parts:

The hypothesis is the part of a conditional statement that follows “if” (when written in if-then form.) It is the given information, or the condition.

The conclusion is the part of an if-then statement that follows “then” (when written in if-then form.) It is the result of the given information.

Conditional Statement

+ Ex 2: Underline the hypothesis & circle the conclusion.

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

hypothesis conclusion

+Conditional statements can be written in “if-then” form to emphasize which part is the hypothesis and which is the conclusion.

Writing Conditional Statements

Hint: Turn the subject into the hypothesis.

Example 3: Vertical angles are congruent. can be written as...

If two angles are vertical, then they are congruent. Conditional Statement:

Example 4: Seals swim. can be written as...

Conditional Statement: If an animal is a seal, then it swims.

+ If …Then vs. Implies

Two angles are vertical implies they are congruent.

Another way of writing an if-then statement is using the word implies.

If two angles are vertical, then they are congruent.

Example 5: Rewrite the statements in if-then form.

All birds have feathers. a.

Solution:

First, identify the hypothesis and the conclusion. When you rewrite the statement in if-then form, you may need to reword the hypothesis or conclusion.

a. All birds have feathers. If an animal is a bird, then it has feathers.

b. Two angles are supplementary if they are a linear pair.

If two angles are a linear pair, then they are supplementary.

b. Two angles are supplementary if they are a linear pair.

Example 5: Rewrite the statements in if-then form.

Solution:

+ Practice: Rewrite the conditional statement in if-then form.

1. All 90° angles are right angles.

Solution:

If the measure of an angle is 90°, then it is a right angle.

2. 2x + 7 = 1, because x = –3.

Solution:

If x = –3, then 2x + 7 = 1.

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3. When n = 9, n2 = 81.

Solution:

If n = 9, then n2 = 81.

4. Tourists at the Coliseum are in Rome.

Solution:

If tourists are at the Coliseum, then they are in Rome.

Practice: Rewrite the conditional statement in if-then form.

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+ Ex 6: Find a counterexample to prove the statement is false.

If x2 = 81, then x must equal 9.

counterexample: x could be -9

because (-9)2 = 81, but x ≠ 9.

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  Writing the opposite of a statement.

  Ex: negate x=3

x≠3

  Ex: negate t>5

t 5

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  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.

+  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.

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  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.

The original conditional statement & its contrapositive will always have the same meaning.

The converse & inverse of a conditional statement will always have the same meaning.

Example 7: Write four related conditional statements (if-then form, the converse, the inverse, and the contrapositive) of “ Guitar players are musicians.” Decide whether each statement is true or false.

SOLUTION

If-then form: If you are a guitar player, then you are a musician.

Converse: If you are a musician, then you are a guitar player.

True, guitars players are musicians.

False, not all musicians play the guitar.

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Inverse: If you are not a guitar player, then you are not a musician.

Contrapositive: If you are not a musician, then you are not a guitar player.

False, even if you don’t play a guitar, you can still be a musician.

True, a person who is not a musician cannot be a guitar player.

Example 7: Write four related conditional statements

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If a dog is a Great Dane, then it is large.

Converse: If the dog is large, then it is a Great Dane. False. Inverse: If dog is not a Great Dane, then it is not large. False.

Solution:

Contrapositive: If a dog is not large, then it is not a Great Dane. True.

Example 8: Write the converse, the inverse, and the contrapositive of the conditional statement. Tell whether each statement is true or false.

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If a polygon is equilateral, then the polygon is regular.

Converse: If Polygon is regular, then it is equilateral. True

Inverse: If a Polygon is not equilateral, then it is not regular. True

Contrapositive: If a Polygon is not regular, then it is not equilateral. False

Solution:

Example 9: Write the converse, the inverse, and the contrapositive of the conditional statement. Tell whether each statement is true or false.

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A conditional statement and its converse combined to make a biconditional statement, which can also be true or false

  Ex: If it is a line, then it contains two points. If it contains two points, then it is a line.

It is a line if and only if it contains at least two points

Solution:

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

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

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

Example 10: Write definition of perpendicular lines as a biconditional.

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Biconditional: An angle is a right angle if and only if the measure of the angle is 90°.

Solution:

Example 10: Rewrite definition of right angle as a biconditional statement.

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If Mary is in theater class, she will be in the fall play. If Mary is in the fall play, she must be taking theater class.

Biconditional: Mary is in the theater class if and only if she will be in the fall play.

Solution:

Example 10: Rewrite definition of right angle as a biconditional statement.