Conditional Statements A conditional statement has two parts, the hypothesis and the conclusion. Written in if-then form: If it is Saturday, then it is.

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Presentation transcript:

Conditional Statements A conditional statement has two parts, the hypothesis and the conclusion. Written in if-then form: If it is Saturday, then it is the weekend. hypothesis conclusion

Rewriting in If-Then Form You will get $1 off a sandwich if you bring in a coupon.

Rewriting in If-Then Form You will get $1 off a sandwich if you bring in a coupon. Identify the hypothesis and the conclusion.

Rewriting in If-Then Form You will get $1 off a sandwich if you bring in a coupon. Identify the hypothesis and the conclusion.

Rewriting in If-Then Form You will get $1 off a sandwich if you bring in a coupon. Identify the hypothesis and the conclusion.

Rewriting in If-Then Form You will get $1 off a sandwich if you bring in a coupon. Identify the hypothesis and the conclusion. If you bring in a coupon, then you will get $1 off a sandwich.

Rewriting in If-Then Form All geometry teachers are totally awesome.

Rewriting in If-Then Form All geometry teachers are totally awesome. If you are a geometry teacher, then you are totally awesome.

Rewriting in If-Then Form A number divisible by 6 is also divisible by 3.

Rewriting in If-Then Form A number divisible by 6 is also divisible by 3. If a number is divisible by 6, then it is also divisible by 3.

Conditional Statements Conditional statements can be either true or false. For a conditional statement to be true, it must be true for all cases. To show a conditional statement is false, you must provide only a single counterexample.

Writing a Counterexample A counterexample must have a true hypothesis and a false conclusion. If x 2 = 16, then x = 4.

Writing a Counterexample A counterexample must have a true hypothesis and a false conclusion. If x 2 = 16, then x = 4. Let x = –4.

Writing a Counterexample A counterexample must have a true hypothesis and a false conclusion. If x 2 = 16, then x = 4. Let x = –4. The hypothesis is true, because (–4) 2 = 16.

Writing a Counterexample A counterexample must have a true hypothesis and a false conclusion. If x 2 = 16, then x = 4. Let x = –4. The hypothesis is true, because (–4) 2 = 16. However, the conclusion is false, because x ≠ 4.

Writing a Counterexample A counterexample must have a true hypothesis and a false conclusion. If x 2 = 16, then x = 4. Let x = –4. The hypothesis is true, because (–4) 2 = 16. However, the conclusion is false, because x ≠ 4. Therefore, the conditional statement is false.

Three Ways to Alter a Conditional Statement Converse Inverse Contrapositive

Converse of a Statement The converse of a conditional statement is formed by switching the hypothesis and conclusion.

Converse of a Statement The converse of a conditional statement is formed by switching the hypothesis and conclusion. If it is Saturday, then it is the weekend.

Converse of a Statement The converse of a conditional statement is formed by switching the hypothesis and conclusion. If it is Saturday, then it is the weekend. If it is the weekend, then it is Saturday.

Converse of a Statement The converse of a conditional statement is formed by switching the hypothesis and conclusion. If it is Saturday, then it is the weekend. If it is the weekend, then it is Saturday. Notice that a converse is not necessarily true.

Inverse of a Statement The inverse of a conditional statement is formed by negating the hypothesis and conclusion.

Inverse of a Statement The inverse of a conditional statement is formed by negating the hypothesis and conclusion. If it is Saturday, then it is the weekend.

Inverse of a Statement The inverse of a conditional statement is formed by negating the hypothesis and conclusion. If it is Saturday, then it is the weekend. If it is not Saturday, then it is not the weekend.

Inverse of a Statement The inverse of a conditional statement is formed by negating the hypothesis and conclusion. If it is Saturday, then it is the weekend. If it is not Saturday, then it is not the weekend. Notice that an inverse is not necessarily true.

Contrapositive of a Statement The contrapositive of a conditional statement is formed by switching and negating the hypothesis and conclusion.

Contrapositive of a Statement The contrapositive of a conditional statement is formed by switching and negating the hypothesis and conclusion. If it is Saturday, then it is the weekend.

Contrapositive of a Statement The contrapositive of a conditional statement is formed by switching and negating the hypothesis and conclusion. If it is Saturday, then it is the weekend. If it is not the weekend, then it is not Saturday.

Contrapositive of a Statement The contrapositive of a conditional statement is formed by switching and negating the hypothesis and conclusion. If it is Saturday, then it is the weekend. If it is not the weekend, then it is not Saturday. Notice that a contrapositive is equivalent to the original conditional statement.

Postulates In mathematics, a postulate (or axiom) is a rule that is accepted as true without proof. Geometry is called an axiomatic system, because our rules are based off of axioms, which are used to prove theorems.

Point, Line, and Plane Postulates Postulates are statements that are accepted as true. They form the foundation on which other statements are built.

Point, Line, and Plane Postulates Euclid’s Five Postulates: 1. A straight line segment can be drawn joining any two points. 2. Any straight line segment can be drawn indefinitely in a straight line. 3. Given any straight line segment, a circle can be drawn having the segment as radius and one endpoint as center. 4. All right angles are congruent. 5. If two lines are drawn which intersect a third in such a way that the sum of the inner angles on one side is less than two right angles, then the two lines inevitably must intersect each other on that side if extended far enough. #5 is commonly known as the Parallel Postulate.

Definitions Along with postulates, we also use definitions in proofs.

Definitions Along with postulates, we also use definitions in proofs. In mathematics, a definition must be true both “forward” and “backward”. A statement and its converse must be true.

Definitions Perpendicular Lines: Lines that intersect to form a right angle.

Definitions Perpendicular Lines: Lines that intersect to form a right angle. If two lines are perpendicular, then they intersect to form a right angle.

Definitions Perpendicular Lines: Lines that intersect to form a right angle. If two lines are perpendicular, then they intersect to form a right angle.

Definitions Perpendicular Lines: Lines that intersect to form a right angle. If two lines are perpendicular, then they intersect to form a right angle. If two lines intersect to form a right angle, then they are perpendicular.

Biconditional Statements A biconditional statement is a statement that contains the phrase “if and only if”. All definitions are biconditional.

Biconditional Statements Writing a definition as a biconditional. Two lines are perpendicular if and only if they intersect at a right angle.

Biconditional Statements Rewriting a biconditional statement. A biconditional can be written as a conditional statement and its converse.

Biconditional Statements Rewriting a biconditional statement. A biconditional can be written as a conditional statement and its converse. Three lines are coplanar if and only if they lie in the same plane.

Biconditional Statements Rewriting a biconditional statement. A biconditional can be written as a conditional statement and its converse. Three lines are coplanar if and only if they lie in the same plane. If three lines are coplanar, then they lie in the same plane.

Biconditional Statements Rewriting a biconditional statement. A biconditional can be written as a conditional statement and its converse. Three lines are coplanar if and only if they lie in the same plane. If three lines are coplanar, then they lie in the same plane. If three lines lie in the same plane, then they are coplanar.

Biconditional Statements Analyzing a Biconditional Statement X = 3 if and only if x 2 = 9. Is this a biconditional statement? Is the statement true?

Biconditional Statements Analyzing a Biconditional Statement X = 3 if and only if x 2 = 9. Is this a biconditional statement? YES Is the statement true?

Biconditional Statements Analyzing a Biconditional Statement X = 3 if and only if x 2 = 9. Is this a biconditional statement? YES Is the statement true? NO

Questions?

Difference Between “if” and “only if” A figure is a rectangle if it is a square. TRUE A figure is a rectangle only if it is a square. FALSE