Conditional Statement A conditional statement has two parts, a hypothesis and a conclusion. When conditional statements are written in if-then form, the.

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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 11/15/2015Geometry4

Example 1: State the hypothesis and conclusion. If you are 13 years old, then you are a teenager. Hypothesis: You are 13 years old Conclusion: You are a teenager 11/15/2015Geometry5

Example 1: 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 is divisible by 9, then it is divisible by 3. 11/15/2015Geometry6

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 11/15/2015Geometry7

Converse, Inverse and Contrapositive Converse The converse of a conditional is formed by switching the hypothesis and the conclusion. The converse of p → q is q → p 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. 11/15/2015Geometry8

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. 11/15/2015Geometry9

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. 11/15/2015Geometry10

Biconditional Statement p if and only if q p iff. q Biconditional Statement It is Saturday, if and only if I am working at the restaurant. Conditional Statement If it is Saturday, then I am working at the restaurant.

Rewrite the biconditional as conditional statement and its converse. Two angles are supplementary if and only if the sum of their measures is 180°. Conditional: If two angles are supplementary, then the sum of their measures is 180°. Converse: If the sum of two angles measure 180°, then they are supplementary.

Conditional Statement: If it rains, then the game will be cancelled. Write down one of the following. Move to the correct corner: inverse, converse, or contrapositive. If the game is cancelled, then it has rained. If it does not rain, then the game will not be cancelled. If the game is not cancelled, then it has not rained. Converse Inverse Contrapositive

Conditional Statement: If there is snow on the ground, then flowers are not in bloom. Write down one of the following. Move to the correct corner: inverse, converse, or contrapositive. If there is no snow on the ground, then flowers are in bloom. If flowers are not in bloom, then there is snow on the ground. If flowers are in bloom, then there is no snow on the ground. Converse Inverse Contrapositive

Conditional Statement: If two points are collinear, then they lie on the same line. Write down one of the following. Move to the correct corner: inverse, converse, or contrapositive. If two points lie on the same line, then they are collinear. If two points do not lie on the same line, then they are not collinear. If two points are not collinear, then they do not lie on the same line. Contrapositive Converse Inverse