Conditional Statements

1. Conditional Statements Goals Recognize a conditional statement Write the converse, inverse, and conditional statement

2. Recognizing Conditional Statements Conditional Statements If-Then Statements If a number is divisible by both 2 and 3 then it is divisible by 6. HYPOTHESIS CONCLUSION If a polygon has four sides then it is a quadrilateral. If a number greater than two is even, then it is not prime.

3. Recognizing Conditional Statements Conditional statements can be True or False • To show a conditional statement is true, you must present an argument to show true in all cases. • To show conditional statement is false, you only have to have a single counterexample.

4. Recognizing Conditional Statements Example: Write a counterexample: If a number is odd, then it is divisible by 3

5. Recognizing Conditional Statements State the hypothesis and conclusion for each statement. IF two angles are supplementary, THEN the sum of their angles is 180 degrees. Hypothesis Conclusion IF two angles are supplementary, THEN the sum of their angles is 180 degrees.

6. Recognizing Conditional Statements State the hypothesis and conclusion for each statement. Example 2 IF two angles are adjacent, THEN they have a common vertex. Hypothesis: Two angles are adjacent Conclusion: The angles have a common vertex

7. Recognizing Conditional Statements Rewrite in if-then form All monkeys have tails. If an animal is a monkey, then the animal has a tail. Vertical angles are congruent. If two angles are vertical, then they are congruent.

8. Recognizing Conditional Statements The CONVERSE of a conditional statement is formed by interchanging the hypothesis and conclusion.  conditional statement If x – y is positive then x > y . converse If x > y then x – y is positive.

9. Recognizing Conditional Statements Write the converse of the following statements. 1. IF two angles are adjacent, THEN they have a common vertex. CONVERSE - IF two angles have a common vertex, THEN they are adjacent. 2. IF two angles are supplementary, THEN the sum of their angles is 180 degrees. CONVERSE - IF two angles have a sum of 180 degrees, THEN they are supplementary.

10. Recognizing Conditional Statements Given a conditional statement, its INVERSE can be formed by negating both the hypothesis and conclusion. The inverse of a true statement is not necessarily true. Conditional statement: If the angle is 75 degrees, then it is acute. Inverse: If the angle is not 75 degrees, then it is not acute.

11. Recognizing Conditional Statements Example 3 Find the inverse of the following statement. Is it True or False If you have vertical angles, then they are congruent. If angles are not vertical angles, then they are not congruent. False

12. Recognizing Conditional Statements CONTRAPOSITIVE: Formed by negating the hypothesis and conclusion of the converse of the given conditional. When forming a contrapositive of a conditional it may be easier to write the converse first – then negate each part. Example: Statement: If the angle is 75 degrees then it is acute . If an angle is not acute then it is not 75 degrees

13. Recognizing Conditional Statements Example 5: Write the contrapositive of the conditional statement If two angles are vertical, then they are congruent. If two angles are not congruent, then they are not vertical angles

14. Biconditional Statements • Biconditional statement is when the conditional statement and converse are both true. It can be written as an “if and only if” statement. Conditional Statement: If an angle is classified as a right angle, then it measures 90 degrees Is the conditional and converse True? If so, then… • An angle is called a right angle if and only if it measures 90 degrees.

15. Recognizing Conditional Statements

16. Write the converse, inverse, contrapositive, and biconditional statement for the following conditional statement. If a triangle is isosceles, then it has two congruent sides.