10/21/2015Geometry1 Conditional Statements

10/21/2015Geometry2 Goals Recognize and analyze a conditional statement Write postulates about points, lines, and planes using conditional statements

10/21/2015Geometry3 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

10/21/2015Geometry4 Examples If you are 13 years old, then you are a teenager. Hypothesis:  You are 13 years old Conclusion:  You are a teenager

10/21/2015Geometry5 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.

Counterexample Used to show a conditional statement is false. It must keep the hypothesis true, but the conclusion false! It must keep the hypothesis true, but the conclusion false!

10/21/2015Geometry7 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 = -4.  The hypothesis is true, but the conclusion is false. Therefore the conditional statement is false.

10/21/2015Geometry8 Converse The converse of a conditional is formed by switching the hypothesis and the conclusion. The converse of p → q is q → p

Converse Switch the hypothesis & conclusion parts of a conditional statement. Ex: Write the converse of “If you are a brunette, then you have brown hair.” If you have brown hair, then you are a brunette.

10/21/2015Geometry10 Negation The negative of the statement Writing the opposite of a statement. Example: Write the negative of the statement   A is acute   A is not acute ~p represents “not p” or the negation of p

Inverse Negate the hypothesis & conclusion of a conditional statement. Ex: Write the inverse of “If you are a brunette, then you have brown hair.” If you are not a brunette, then you do not have brown hair.

Contrapositive Negate, then switch the hypothesis & conclusion of a conditional statement. Ex: Write the contrapositive of “If you are a brunette, then you have brown hair.” If you do not have brown hair, then you are not a brunette.

10/21/2015Geometry13 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.

10/21/2015Geometry14 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.

10/21/2015Geometry15 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.

10/21/2015Geometry16 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

10/21/2015Geometry17 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