2.1, 2.2 and 5.4: Statements and Reasoning

Conditional is an if-then statement that contains two parts. The part following the if is the Hypothesis. The part following then is the Conclusion. Example 1: If you live in a country that borders the northern United States, then you live in Canada. (2.1) Conditional Statements Hypothesis : You live in a country that borders the northern United States. Conclusion : You live in Canada.

Conditional is an if-then statement that contains two parts. The part following the if is the Hypothesis. The part following then is the Conclusion. Example 2: If two lines are parallel, then the lines are coplanar. (2.1) Conditional Statements Hypothesis: Two lines are parallel. Conclusion: The lines are coplanar.

Conditional is an if-then statement that contains two parts. The part following the if is the Hypothesis. The part following then is the Conclusion. Example 3: Write the statement as a conditional. An acute angle measures less than 90 degrees. (2.1) Conditional Statements Hypothesis : An angle is acute First part of the Conditional: If an angle is acute Conclusion : It measures less than 90 degrees Second part of the Conditional: then it measures less than 90 degrees Conditional Statement: If an angle is acute, then it measures less than 90 degrees.

Conditional is an if-then statement that contains two parts. The part following the if is the Hypothesis. The part following then is the Conclusion. Example 4: Write a counterexample to prove the conditional false. If x 2 > 0, then x > 0. The counterexample for the conditional is when the hypothesis is true but the conclusion is false. (2.1) Conditional Statements

The converse of a conditional switches the hypothesis and the conclusion. Example 1: Writing the converse of the conditional. Conditional: If two lines intersect to form right angles, then they are perpendicular. (2.1) Converse Converse: If two lines are perpendicular, then they intersect to form right angles.

Converse of a conditional switches the hypothesis and the conclusion. Example 2: Writing the converse of the conditional. Conditional: If x = 9, then x + 3 = 12 Converse: (2.1) Converse If x + 3 = 12, then x = 9

Converse of a conditional switches the hypothesis and the conclusion. Example 3: Write the converse of the conditional and determine the truth value of each. Conditional: If a = 5, then a 2 = 25 Converse: (2.1) Converse If a 2 = 25, then a = 5 *Sometimes the converse is false!!!

Biconditionals When a conditional and its converse are true, you can combine them as a true biconditional. You can combine them by using the phrase if and only if. Example 1: Consider the conditional. Write its converse. If they are both true, combine the statements as a biconditional. Conditional: If two angles have the same measure, then the angles are congruent. Converse: Biconditional: (2.2) Biconditionals If two angles are congruent, then the angles have the same measure. Two angles have the same measure if and only if the angles are congruent.

Biconditionals When a conditional and its converse are true, you can combine them as a true biconditional. You can combine them by using the phrase if and only if. Example 2: Write the two statements that form the biconditional. Biconditional: You live in Washington, D.C., if and only if you live in the capital of the United States. (2.2) Biconditionals Conditional: If you live in Washington, D.C., then you live in the capital of the United States. Converse: If you live in the capital of the United States, then you live in Washington, D.C.

Example: Negate the following statement Original statement: Two angles are congruent. (5.4) Negation The negation of a statement has the opposite meaning of the original statement. Negation: Two angles are not congruent.

Example: Find the inverse of the following conditional statement Conditional: If a figure is a square, then it is a rectangle. (5.4) Inverse The inverse of a conditional statement negates both the hypothesis and the conclusion. Inverse: If a figure is not a square, then it is not a rectangle.

Example: Find the contrapositive of the following conditional. Conditional: If a figure is a square, then it is a rectangle. (5.4) Contrapositive The contrapositive of a conditional statement switches the hypothesis and the conclusion and negates both. Contrapositive: If a figure is not a rectangle, then it is not a square.

Example: Write the (a) the inverse and (b) contrapositive of Maya Angelou’s statement. (5.4) Contrapositive The contrapositive of a conditional statement switches the the hypothesis and the conclusion and negates both. “If you don’t stand for something, then you’ll fall for anything.”

Example: Identify the two statements that contradict each other. I. II. III. (5.4) Identifying Contradictions Two segments cannot be parallel and perpendicular. So I and III contradict each other. Two segments can be parallel and congruent. So I and II do not contradict each other. Two segments can be congruent and perpendicular. So II and III do not contradict each other.

2.1, 2.2 and 5.4: Statements and Reasoning HOMEWORK: Page 83 #3, even, even; Page 90 #2, 6, even, Challenge: even Page 283 #2-6 even, 7-9 all, all TERMS: Biconditional, Conclusion, Conditional, Converse, Hypothesis, *Truth Value: true or false