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Section 1.4 If-Then Statements and Postulates

Section 1.4 If-Then Statements and Postulates. Objectives-What we’ll learn. Recognize and analyze a conditional statement Write postulates about points, lines, and planes using conditional statements. Postulate vs. Theorem. A postulate is a statement that is assumed true without proof.

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Section 1.4 If-Then Statements and Postulates

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  1. Section 1.4 If-Then Statementsand Postulates Geometry

  2. Objectives-What we’ll learn • Recognize and analyze a conditional statement • Write postulates about points, lines, and planes using conditional statements Geometry

  3. Postulate vs. Theorem A postulate is a statement that is assumed true without proof. A theorem is a true statement that can be proven.

  4. 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

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

  6. Rewrite in the if-then form (Conditional Statement) • 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 is divisible by 9, then it is divisible by 3. Geometry

  7. Rewrite in the if-then form (Conditional Statement) • Two lines intersect at a point. • If two lines intersect, then they intersect at a point. • Three non-collinear points determine a plane. • If there are three non-collinear points, then they determine a plane. Geometry

  8. 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

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

  10. Rewrite in the Converse form. • If you are 13 years old, then you are a teenager. If you are a teenager, then you are 13 years old. • If a number divisible by 9, then it is also divisible by 3 If a number is divisible by 3, then it is divisible by 9.

  11. Rewrite in the Converse form. • If two angles are vertical angles, then they are congruent. If two angles are congruent, then they are vertical angles. • If a quadrilateral has 4 right angles, then it is a rectangle. If a quadrilateral is a rectangle, then it has 4 right angles.

  12. Point, Line, and Plane Postulates • Postulate 1-1: Through any two points there exists exactly one line • Postulate 1-2: Through any three noncollinear points there exists exactly one plane • Postulate 1-3: A line contains at least two points • Postulate 1-4: 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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