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Geometry

Geometry. 2.1 Conditional Statements. VOCAB. Conditional Statement – A type of logical statement that has two parts, a hypothesis and a conclusion

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Geometry

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  1. Geometry 2.1 Conditional Statements

  2. VOCAB • Conditional Statement – A type of logical statement that has two parts, a hypothesis and a conclusion • If-Then Form – The form of a conditional statement that uses the words “if” and “then.” The “if” part contains the hypothesis and the “then” part contains the conclusion. • Hypothesis – The “if” part of a conditional statement • Conclusion – The “then” part of a conditional statement

  3. If-Then Form If it is noon in Georgia, then it is 9 a.m. in California.

  4. Rewriting in If-Then Form Rewrite the conditional statement in if-then form. • Two points are collinear if they lie on the same line. • All sharks have a boneless skeleton • A number divisible by 9 is also divisible by 3. • All mammals breathe oxygen.

  5. Rewriting in If-Then Form • Two points are collinear if they lie on the same line. If two points lie on the same line, then they are collinear. • All sharks have a boneless skeleton If a fish is a shark, then it has a boneless skeleton. • A number divisible by 9 is also divisible by 3. If a number is divisible by 9, then it is divisible by 3. • All mammals breathe oxygen. If an animal is a mammal, then it breathes oxygen.

  6. Writing a Counterexample Write a counterexample to show that the following conditional statement is false. • If x squared equals 16, then x equals 4. • If a number is odd, then it is divisible by 3.

  7. Other Statements • Converse – The statement formed by switching the hypothesis and conclusion of a conditional statement. • Inverse – The statement formed when you negate the hypothesis and conclusion of a conditional statement. (negation – the negative of a statement) • Contrapositive– The statement formed when you negate the hypothesis and conclusion of the converse of a conditional statement

  8. Example

  9. POSTULATE 5 – Through any two points there exists exactly one line.

  10. POSTULATE 6 – A line contains at least two points.

  11. POSTULATE 7 – If two lines intersect, then their intersection is exactly one point.

  12. POSTULATE 8 – Through any three noncollinear points there exists exactly one point.

  13. POSTULATE 9 – A plane contains at least three noncollinear points.

  14. POSTULATE 10 – If two points lie in a plane, then the line containing them lies in the plane.

  15. POSTULATE 11 – If two planes intersect, then their intersection is a line.

  16. Rewriting a Postulate Rewrite Postulate 5 in if-then form and then write its inverse, converse, and contrapositive. • Conditional – • Inverse – • Converse – • Contrapositive

  17. Rewriting a Postulate • Conditional – If two points are distinct, then there is exactly one line that passes through them. • Inverse – If two points are not distinct, then it is not true that there is exactly one line that passes through them. • Converse – If exactly one line passes through two points, then the two points are distinct. • Contrapositive – If it is not true that exactly one line passes through two points, then the two points are not distinct.

  18. Using Postulates and Counterexamples Decide whether the statement is true or false. If it is false, give a counterexample. • A line can be in more than one plane. • Four noncollinear points are always coplanar. • Two intersecting lines can be noncoplanar. • A line can contain more than two points.

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