1. 2.1 Conditional Statements GOAL 1 Recognize and analyze a conditional statement. GOAL 2 Write postulates about points, lines, and planes using conditional statements. What you should learn Point, line, and plane postulates help you analyze real-life objects, such as a research buggy (pg. 71). Why you should learn it

2. GOAL 1 RECOGNIZING CONDITIONAL STATEMENTS VOCABULARY A conditional statement has two parts, a __________ and a __________. If the statement is written in if-then form, the “if” part contains the __________ and the “then” part contains the __________. 2.1 Conditional Statements hypothesis conclusion hypothesis conclusion If HYPOTHESIS, then CONCLUSION. EXAMPLE 1

3. Extra Example 1 Rewrite in if-then form: All mammals breathe oxygen. Hypothesis: Conclusion: An animal is a mammal. It breathes oxygen. If an animal is a mammal, then it breathes oxygen. If-then form:

4. EXAMPLE 2 TRUE OR FALSE? For a conditional statement to be TRUE, the conclusion must be true for all cases that fulfill the hypothesis. It will be YOUR JOB to show this. For a conditional statement to be FALSE, only one counterexample is necessary.

5. Extra Example 2 Write a counterexample: If a number is odd, then it is divisible by 3. Sample answer: 7 is odd and 7 is not divisible by 3.

6. Converse:The hypothesis and conclusion are switched. EXAMPLE 3 CAUTION: When writing negations, do not be more specific than the original statement. The negation of “It is raining” is “It is not raining,” NOT “It is sunny.” (It could be night, foggy, etc.). NEGATIONS Inverse: Contrapositive: Both hypothesis and conclusion are negated. The hypothesis and conclusion of a converse are negated.

7. EQUIVALENT STATEMENTS Two statements are either both true or both false. A conditional is equivalent to its contrapositive, and the inverse and converse of any conditional statement are equivalent. EXAMPLE 4

8. Extra Example 4 Write the a) inverse, b) converse, and c) contrapositive of the statement. If the amount of available food increases, the deer population increases. Inverse: Converse: Contrapositive: If the amount of available food does not increase, the deer population does not increase. If the deer population increases, the amount of available food increases. If the deer population does not increase, the amount of available food does not increase. Hint: Identify the hypothesis and conclusion first.

9. Checkpoint 1.Rewrite in if-then form: All monkeys have tails. 2.Write a counterexample: If a number is divisible by 2, then it is divisible by 4. 3. Write the inverse, converse, and contrapositive of the statement: If an animal is a fish, then it can swim. If an animal is a monkey, then it has a tail. Sample: 14 is divisible by 2 but not divisible by 4. Inverse:If an animal is not a fish, then it cannot swim. Converse:If an animal can swim, then it is a fish. Contrapositive:If an animal cannot swim, then it is not a fish.

10. GOAL 2 USING POINT, LINE, AND PLANE POSTULATES 2.1 Conditional Statements Study the postulates on page 73, and be sure you understand what they mean. Example 5 will help you. EXAMPLE 5

11. Extra Example 5 EXAMPLE 6 P5: There is exactly one line (m) through A and B. P6: m contains at least two points (A and B). P7: m and n intersect at C. P8: Q passes through A, B, and D. P9: Q contains at least A, B, and D. P10: A and B lie in Q. So m, which contains A and B, also lies in Q. P11: P and Q intersect in line n. A D B C m n P Q Give examples of Postulates 5-11.

12. Extra Example 6 Rewrite Postulate 6 in if-then form, then write its inverse, converse, and contrapositive. If-then form: Inverse: Converse: Contrapositive: If a figure is a line, then it contains at least two points. If a figure is not a line, then it does not contain at least two points. If a figure contains at least two points, then it is a line. If a figure does not contain at least two points, then it is not a line. EXAMPLE 7

13. Extra Example 7 Decide if the statement is true or false. If it is false, give a counterexample. Three points are always contained in a line. Sample answer: FALSE A B C

14. Checkpoint 1.Write the inverse, converse, and contrapositive of Postulate 8. 2.Decide whether the statement is true or false. If it is false, give a counterexample. A line can contain more than two points. Inverse: If three noncollinear points are not distinct, then it is not true that there is exactly one plane that passes through them. Converse: If exactly one plane passes through three noncollinear points, then the three points are distinct. Contrapositive: If it is not true that exactly one plane passes through three noncollinear points, then the three points are not distinct. TRUE

15. Vocabulary Check Conditional Statement If-then form Hypothesis Conclusion Converse Inverse Contrapositive Negation Equivalent Statements