1. Section 2.2 Conditional Statements 1 Goals Recognize and analyze a conditional statement Write postulates about points, lines, and planes using conditional statements

2. Conditional Statement A conditional statement has two parts, When conditional statements are written in if- then form, the part after the “if” is the __________, and the part after the “then” is the __________. Symbolic notation: p → q 2

3. Examples If you are 13 years old, then you are a teenager. Hypothesis: Conclusion: 3 If there is snow on the ground, then it is cold. Hypothesis: Conclusion:

4. Rewrite in the if-then form  An angle which measures 45  is acute.  A number divisible by 9 is also divisible by 3 4 It is time for lunch if it is noon. A triangle which is equilateral is also isosceles.

5. 5  True Statement  Statement which is always true  False Statement  Statement which is not always true In order to prove something is false, we only need to show one example where it is false. We call that example a ___________. Counterexample – an example which proves a statement is false

6. Writing a Counterexample Write a counterexample to show that the following conditional statement is false – If x 2 = 16, then x = 4. – As a counterexample, let x =. The hypothesis is _____, but the conclusion is _____. Therefore the conditional statement is _____. 6

7. Converse The converse of a conditional is formed by switching the hypothesis and the conclusion. The converse of p → q is q → p 7 Write the converse of the following conditional statements: Conditional: If I play football, then I am an athlete. Converse: Conditional: If two segments are congruent, then they have the same length. Converse:

8. Negation The negative of the statement Example: Write the negative of the statement –  A is acute –  A is ____ acute ~p represents “not p” or the negation of p 8

9. Inverse Statements Inverse – Negate the hypothesis and the conclusion of the conditional statement – The inverse of p → q, is ~p → ~q Write the inverse of the following conditionals: Conditional: If an angle measures 45 , then it is an acute angle. Inverse: Conditional: If two segments are congruent, then they have the same length. Inverse:

10. Contrapositive Statements 10 Contrapositive Contrapositive Negate the hypothesis and the conclusion of the converse Negate the hypothesis and the conclusion of the converse The contrapositive of p → q, is ~q → ~p. The contrapositive of p → q, is ~q → ~p. Write the contrapositive of the following conditionals: Conditional: If an angle measures 45 , then it is an acute angle. Contrapositive: Conditional: If two segments are congruent, then they have the same length. Contrapositive:

11. Example Write the (a) converse, (b) inverse, and (c) contrapositive of the statement. – If two angles are vertical, then the angles are congruent. (a) Converse: (b) Inverse: (c) Contrapositive: 11

12. Equivalent Statements When 2 statements are both true or both false A conditional statement is equivalent to its contrapositive. The converse statement is equivalent to the inverse statement 12 Conditional Converse Inverse Contrapositive

13. 13 Definitions: All definitions can be read both forwards and backwards. If two lines are perpendicular, then they intersect to form a right angle. Forward: Backward: If two angles are complementary, then their sum is 90  Forward: Backward:

14. 14 Conditional: If two angles are supplementary, then the sum of the two angles is 180  Converse: Conditional: Converse: If the sum of two angles is 180 , then the two angles are supplementary.

15. 15 Biconditional Statement Is a statement that contains the phrase “if and only if” Is a statement that contains the phrase “if and only if” This is equivalent to writing a conditional statement and its converse This is equivalent to writing a conditional statement and its converse Can be either true or false Can be either true or false To be true, both the conditional and converse must be true To be true, both the conditional and converse must be true Symbolically: Symbolically: p ↔ q Biconditional Statements Biconditional Two angles are supplementary if and only if their sum is 180 .

16. 16 Writing a Postulate as a Biconditional Postulate: (Conditional) If P is in the interior of  RST, then  RSP +  PST   RST Write the converse and decide if it is true. Converse: If  RSP +  PST   RST, then P is in the interior of  RST. Combine it with the postulate to form a true biconditional. Biconditional: