Section 2.1 Conditional Statements

1. Section 2.1 Conditional Statements Geometry

2. Goals • Recognize and analyze a conditional statement • Write postulates about points, lines, and planes using conditional statements Geometry

3. Conditional Statement • A conditional statement has two parts, a hypothesis and a conclusion. • When conditional statements are written in if-then form, the part after the “if” is the hypothesis, and the part after the “then” is the conclusion. • p → q Geometry

4. Examples • If you are 13 years old, then you are a teenager. • Hypothesis: • You are 13 years old • Conclusion: • You are a teenager Geometry

5. Rewrite in the if-then form • All mammals breathe oxygen • If an animal is a mammal, then it breathes oxygen. • A number divisible by 9 is also divisible by 3 • If a number s divisible by 9, then it is divisible by 3. Geometry

6. Writing a Counterexample • Write a counterexample to show that the following conditional statement is false • If x2 = 16, then x = 4. • As a counterexample, let x = -4. • The hypothesis is true, but the conclusion is false. Therefore the conditional statement is false. Geometry

7. Converse • The converse of a conditional is formed by switching the hypothesis and the conclusion. • The converse of p → q is q → p Geometry

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

9. Inverse and Contrapositive • Inverse • Negate the hypothesis and the conclusion • The inverse of p → q, is ~p → ~q • Contrapositive • Negate the hypothesis and the conclusion of the converse • The contrapositive of p → q, is ~q → ~p. Geometry

10. Example • Write the (a) inverse, (b) converse, and (c) contrapositive of the statement. • If two angles are vertical, then the angles are congruent. • (a) Inverse: If 2 angles are not vertical, then they are not congruent. • (b) Converse: If 2 angles are congruent, then they are vertical. • (c) Contrapositive: If 2 angles are not congruent, then they are not vertical. Geometry

11. Equivalent Statements • When 2 statements are both true or both false • A conditional statement is equivalent to its contrapositive. • The inverse and the converse of any conditional are equivalent. Geometry

12. Point, Line, and Plane Postulates • Postulate 5: Through any two points there exists exactly one line • Postulate 6: A line contains at least two points • Postulate 7: If 2 lines intersect, then their intersection is exactly one point • Postulate 8: Through any three noncollinear points there exists exactly one plane Geometry

13. Postulate 9: A plane contains at least three noncollinear points • Postulate 10: If two points lie in a plane, then the line containing them lies in the plane • Postulate 11: If two planes intersect, then their intersection is a line Geometry