Conditional & Biconditional Statements

1. Conditional & Biconditional Statements Chapter 2 Sections 1 & 2

2. Hypothesis Conclusion Symbolic Notation p  q (If p then q) q  p (If q then p) Conditional Statements an “if-then” statement IF <Hypothesis> then <conclusion> Examples: If it is a nice day then I will go to the park

3. Converse The converse of a conditional switches the hypothesis and the conclusion. Example Conditional: If 2 lines intersect to form right angles then they are perpendicular. Converse: If 2 lines are perpendicular then they intersect to form right angles.

4. Truth Value A conditional statement can have a truth value of true or false. To show a conditional is TRUE Show that every time the hypothesis is true, the conclusion is true To show a conditional is FALSE Find a counterexample in which the hypothesis is TRUE and the conclusion is FALSE

5. A “good” definition: • Uses clearly understandable terms • Precise. (avoids words such as “large, sort of, and some”) • Reversable. (must be able to be written as a TRUE converse).

6. Example Conditional: If a figure is a square then it has 4 sides. True or False? Converse: If a figure has 4 sides then it is a square. True or False?

7. Practice Write the converse of the conditional statement. a) If an angle has less than 90 degrees, then it is an acute angle. If an angle is acute, then it measures less than 90 degrees. b) If a figure has four congruent sides, then it is a square. If a figure is a square, then it has four congruent sides.

8. Biconditional Statement the combination of a conditional statement and its converse (as long as both statements are true). statements are combined using the phrase “if and only if” (iff) Example: Two angles have the same measure if and only if the angles are congruent.

9. Symbolic Form

10. Example Write the biconditional: Conditional: If three points are collinear, then they lie on the same line. TRUE Converse: If three points lie on the same line, then they are collinear. TRUE Biconditional: Three points are collinear iff they lie on the same line.

11. Example Separate the biconditional into two conditional statements. Biconditional: A number is divisible by 3 if and only if the sum of its digits is divisible by 3. Conditional: If a number is divisible by 3, then the sum of its digits is divisible by 3. Conditional (converse): If the sum of a number’s digits is divisible by 3, then the number is divisible by 3.

12. Remember: HW is due next class! NO make-up work is allowed. Workbook Pages 15 – 18

13. Deductive Reasoning& Reasoning in Algebra Chapter 2 Sections 3 & 4

14. Review Conditional: If three points are collinear, then they lie on the same line. Converse: If three points lie on the same line, then they are collinear. Biconditional: Three points are collinear if and only if they lie on the same line.

15. Deductive Reasoning the process of reasoning logically from accepted facts, definitions and properties to a conclusion. Law of Detachment If a conditional statement is true and the hypothesis is true, then the conclusion is true. Example If a car has a dead battery, then the car will not start. Jim’s car has a dead battery. Conclusion: Jim’s car will not start.

16. Law of Syllogism Stating a conclusion from two true conditional statements when the conclusion of one statement is the hypothesis of the other statement. If p→q and q→r, then p→r. Example If you spend time with friends, then you enjoy yourself. If you enjoy yourself, then your time is well spent. Conclusion: If you spend time with friends, then your time is well spent.

17. Practice Use the Law of Detachment to draw a conclusion. If two lines are parallel, then they do not intersect. Line f is parallel to line m. Lines f & m do not intersect. Use the Law of Syllogism to draw a conclusion. If two planes intersect, then they intersect in a line. If two planes are not parallel, then they intersect. If two planes are not parallel, then they intersect in a line.

18. Properties of Equality Addition Property If a = b, then a + c = b + c Ex. x – 2 = 4 Given x – 2 + 2 = 4 + 2 Addition Prop. x = 6 Simplify

19. Properties of Equality Multiplication Property If a = b, then a • c = b • c Ex. x/2 = 4 Given 2 •x/2 = 4 • 2 Multiplication Prop. x = 8 Simplify

20. Properties of Equality Division Property If a = b, then a/c = b/c Ex. 2x = 4 Given 2x / 2 = 4 / 2 Division Prop. x = 2 Simplify

21. Properties of Equality Substitution Property If a = b, then b can replace a in any expression Ex. Let x = 2 and y = 3x + 4 y = 3x + 4 Given y = 3 (2) + 4 Substitution Prop. y = 6 + 4 = 10 Simplify

22. Properties of Equality Distributive Property a(b + c) = ab + ac Ex. 2(x + 4) Given 2x + 8 Distributive Prop.

23. Properties of Equality Reflexive Property a = a Symmetric Property If a = b, then b = a Transitive Property If a = b and b = c, then a = c

24. B A O C Ex. Solve for x and justify each step. Given: Angle Addition Prop Substitution Prop Simplify Subtraction Prop Division Prop

25. Ex. Solve for x and justify each step. G E F D Angle Addition Prop Substitution Prop Simplify Subtraction Prop Division Prop

26. Ex. Solve for x and justify each step. Given: C B A Segment Add. Prop. Substitution Prop Distributive Prop Simplify Subtraction Prop Division Prop

27. Remember: HW is due next class! NO make-up work is allowed. Workbook Pages 19 – 22

28. Proving Angles Congruent Chapter 2 Section 5

29. Review Addition Property Reflexive Property If a = b, then a + c = b + c a = a Multiplication Property Symmetric Property If a = b, then b = a If a = b, then a • c = b • c Division Property Transitive Property If a = b, then a/c = b/c If a = b and b = c, then a = c Substitution Property If a = b, then b can replace a in any expression Distributive Property a(b + c) = ab + ac

30. Types of Angles Acute Obtuse Right Straight

31. Types of Angles Vertical Adjacent 3 1 2 1 4 2 1 and 2 are vertical 3 and 4 are vertical 1 and 2 are adjacent

32. Types of Angles Supplementary Complementary 2 1 2 1 1 and 2 are complementary 1 and 2 are supplementary

33. Example: 90° Solve for x. 90° 5 5

34. Example 90° Identify the angle pairs. 1 2 3 5 4 180° 180° Complementary Angles? Supplementary Angles? Vertical Angles?

35. Vertical Angle Theorem Vertical angles are congruent 3 1 2 4

36. Example Solve for x. -4x -4x -3 -3 -2 -2

37. Practice Solve for x and y 180° -x -x 3 3

38. Remember: HW is due next class! NO make-up work is allowed. Workbook Pages 23 – 24