1. Section 1.4 If-Then Statements and Postulates
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Geometry

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

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