2.1 Conditional Statements

1. 2.1 Conditional Statements

2. Objectives/Assignment • Recognize and analyze a conditional statement • Write postulates about points, lines, and planes using conditional statements • Assignment: 2-48 even

3. Goal 1: Recognizing Conditional Statement • A logical statement with 2 parts • 2 parts are called the hypothesis & conclusion • Can be written in “if-then” form; such as, “If…, then…” • Hypothesis is the part after the word “If” • Conclusion is the part after the word “then”

4. Ex: Underline the hypothesis & circle the conclusion. • If you are a brunette, then you have brown hair. hypothesis conclusion

5. Ex: Rewrite the statement in “if-then” form • Vertical angles are congruent. If there are 2 vertical angles, then they are congruent. If 2 angles are vertical, then they are congruent.

6. Ex: Rewrite the statement in “if-then” form • An object weighs one ton if it weighs 2000 lbs. If an object weighs 2000 lbs, then it weighs one ton.

7. 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! • It must keep the hypothesis true, but the conclusion false!

8. Ex: Find a counterexample to prove the statement is false. • If x2=81, then x must equal 9. counterexample: x could be -9 because (-9)2=81, but x≠9.

9. Negation • Writing the opposite of a statement. • Ex: negate x=3 x≠3 • Ex: negate t>5 t 5

10. 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.

11. 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.

12. Contrapositive • Negate, then switch the hypothesis & conclusion of a conditional statement (negate the converse) • 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.

13. Equivalent Statements Equivalent Statements – both are true or both are false. The original conditional statement & its contrapositive will always have the same meaning. The converse & inverse of a conditional statement will always have the same meaning.

14. Goal 2: Using Point, Line, and Plane Postulates Postulate 5: Through any 2 points, there exists exactly one line. Postulate 6: A line contains at least 2 points. Postulate 7: If two lines intersect, then their intersection is exactly one point. Postulate 8: Through any 3 noncollinear points, there exists exactly one plane. Postulate 9: A plane contains at least 3 noncollinear points. Postulate 10: If 2 points lie in a plane, then the line containing them lies in the plane. Postulate 11: If 2 planes intersect, then their intersection is a line.

15. Identifying Postulates There is exactly one line (line n) that passes through the points A and B. Line n contains at least two points, A and B. Lines m and n intersect at point A. Plane P passes through the noncollinear points A, B, and C. Plane P contains at least three noncollinear points, A, B, and C. Points A and B lie in plane P. So, line n, which contains points A and B, also lies in plane P. Planes P and Q intersect. So, they intersect in a line, labeled in the diagram as line m. Postulate 5 Postulate 6 Postulate 7 Postulate 8 Postulate 9 Postulate 10 Postulate 11