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Section 2.2 If-Then Statements and Postulates

Section 2.2 If-Then Statements and Postulates. Goals. Recognize and analyze a conditional statement Write postulates about points, lines, and planes using conditional statements. Conditional Statement. A conditional statement has two parts, a hypothesis and a conclusion .

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Section 2.2 If-Then Statements and Postulates

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  1. Section 2.2 If-Then Statementsand Postulates 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 represents “if p then 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 • Inverse • Negate the hypothesis and the conclusion • The inverse of p → q, is ~p → ~q Geometry

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

  11. 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

  12. Point, Line, and Plane Postulates • Postulate 2-1: Through any two points there exists exactly one line • Postulate 2-2: Through any three noncollinear points there exists exactly one plane • Postulate 2-3: A line contains at least two points • Postulate 2-4: A plane contains at least three points not on the same line Geometry

  13. Postulate 2-5: If two points lie in a plane, then the line containing them lies in the plane • Postulate 2-6: If two planes intersect, then their intersection is a line Geometry

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