2.1 Conditional Statements

1. 2.1 Conditional Statements Day 1 Part 1 CA Standards 1.0, 3.0

2. Warmup • State whether each sentence is true or false. • If you live in Los Angeles, then you live in California. • If you live in California, then you live in Los Angeles. • If today is Wednesday, then tomorrow is Thursday. • If tomorrow is Thursday, then today is Wednesday. True False True True

3. Conditional Statement • Conditional statement has two parts, hypothesis and a conclusion. • If _____________, then____________. hypothesis conclusion

4. Rewrite in If-Then form • A number divisible by 9 is also divisible by 3. • If a number is divisible by 9, then it is divisible by 3. • Two points are collinear if they lie on the same line. • If two points lie on the same line, then they are collinear.

5. Writing a Counterexample • Write a counterexample to show that the following conditional statement is false. • If x2 = 16, then x = 4.

6. Converse • Two points are collinear if they lie on the same line. • If two points are collinear, then they lie on the same line. • If two points lie on the same line, then they are collinear. Conditional Statement Converse

7. A statement can be altered by negation, that is, by writing the negative of the statement. • Statement: m<A = 30° • Negation: m < A ≠ 30° • Statement: <A is acute. • Negation: <A is not acute.

8. Inverse • If two points lie on the same line, then they are collinear. • If two points do not lie on the same line, then they are not collinear. Conditional Inverse

9. Contrapositive • If two points lie on the same line, then they are collinear. • If two points are not collinear, then they do not lie on the same line. Conditional Contrapositive

10. When two statements are both true or both false, they are called equivalent statements. • A conditional statement is equivalent to its contrapositive. • The inverse and converse of any conditional statement are equivalent.

11. Write the • a) inverse • b) converse • c) contrapositive If there is snow on the ground, then flowers are not in bloom. • If there is no snow on the ground, then flowers are in bloom. • If flowers are not in bloom, then there is snow on the ground. • If flowers are in bloom, then there is no snow on the ground.

12. Point, Line, and Plane Postulates • Postulate 5 • Through any two points there exists exactly one line. • Postulate 6 • A line contains at least two points. • Postulate 7 • If two lines intersect, then their intersection is exactly one point.

13. Postulate 8 • Through any three noncollinear points there exists exactly one plane. • Postulate 9 • A plane contains at least three noncollinear points. • Postulate 10 • If two points lie in a plane, then the line containing them lies in the plane.

14. Postulate 11 • If two planes intersect, then their intersection is a line.

15. Review • Write the converse of the conditional statement. • If x = 3, then y = 7. • If Carrie joins the softball team, then Mary will join. • If two angels are vertical, then their measures are equal.

16. 2.2 Definitions and Biconditional Statements Day 1 Part 2 CA Standards 1.0, 3.0

17. Definition • Two lines are called perpendicular lines if they intersect to form a right angle. • A line perpendicular to a plane is a line that intersects the plane in a point and is perpendicular to every line in the plane that intersects it.

18. Exercise • Decide whether each statement about the diagram is true. Explain your answer using the definitions you have learned. • Points D, X, and B are collinear. • AC is perpendicular to DB. • <AXB is adjacent to <CXD. . A . . D X B . C

19. Biconditional Statement • Biconditional Statement • It is Saturday, only if I am working at the restaurant. • Conditional Statement • If it is Saturday, then I am working at the restaurant.

20. Consider the following statement x = 3 if and only if x2 = 9. • Is this a biconditional statement? Yes • Is the statement true? No, because x also can be -3.

21. Rewrite the biconditional as conditional statement and its converse. • Two angles are supplementary if and only if the sum of their measures is 180°. • Conditional: If two angles are supplementary, then the sum of their measures is 180°. • Converse: If the sum of two angles measure 180°, then they are supplementary.

22. State a counterexample that demonstrates that the converse of the statement is false. • If three points are collinear, then they are coplanar. • If an angle measures 48°, then it is acute.

23. Pg. 75 # 4 – 38 even • Handout 2.2

24. 2.3 Deductive Reasoning Day 2 Part 1 CA Standards 1.0, 3.0

25. Warmup • Rewrite the true statement in if-then form and write the converse. If the converse is true, combine it with the if-then statement to form a true biconditional statement. • The perimeter of a triangle is the sum of the lengths of its sides • Two angles measure 42 and 48 form a complementary angle.

26. Symbolic Notations implies • Conditional Statement ( p q) • If the sun is out, then the weather is good. • Converse Statement ( q p) • If the weather is good, then the sun is out. p q q p

27. Exercise • Let p be “the values of x is -5” and let q be “the absolute value of x is 5”. • Write p q in words. • Write q p in words. If the values of x is -5, then the absolute value of x is 5. If the absolute value of x is 5, then the values of x is -5.

28. Symbolic Notations • Conditional Statement ( p q) • If the sun is out, then the weather is good. • Inverse Statement (~ p ~q) • symbol of negation (~) • If the sun is not out, then the weather is not good. p q ~q ~p

29. In summary…

30. Law of Detachment • Law of Detachment • If p q is a true conditional statement and p is true, then q is true. • Law of Syllogism • If p q and q r are true conditional statements, then p r is true.

31. Law of Syllogism • If a bird is the fastest bird on land, then it is the largest of all birds. • If a bird is the largest of all birds, then it is an ostrich. • Combine two conditional statements together • If a bird is the fastest bird on land, then it is an ostrich.

32. Review • Fill in the box with appropriate symbols and examples.

33. Review • Rewrite the conditional statement in if-then form. • It must be true if you read it in a newspaper. If you read it in a newspaper, then it must be true.

34. Review • Write the inverse, converse, and contrapositive of the conditional statement. • If you are indoors, then you are not caught in a rainstorm. Inverse: If you are not indoors, then you are caught in a rainstorm. Converse: If you are not caught in a rainstorm, then you are indoors. Contrapositive: If you are caught in a rainstorm, then you are not indoors.

35. Review • Using p, q, r and s below, write the symbolic statements in words. • p: we go shopping. • q: we need a shopping list • r: we stop at the bank • s: we see our friends. • p q • ~p ~s If we go shopping, then we need a shopping list. If we do not go shopping, then we do not see our friends.

36. 2.4 Reasoning with Properties from Algebra Day 2 Part 2 CA Standard 3.0

37. Algebraic Properties of Equality • Addition property: If a=b, then a+c = b+c. • Subtraction property: If a=b, then a-c = b-c. • Multiplication property: If a=b, then ac = bc. • Division property: If a=b, and c≠0, then a/c = b/c.

38. Solve • Solve 5x – 18 = 3x + 2 and explain each step in writing. 5x – 18 = 3x + 2 2x – 18 = 2 2x = 20 x = 10 Subtraction p. of e. Addition p. of e. Division p. of e.

39. More properties of equality • Reflexive property: For any real number a, a=a. • Symmetric property: If a=b, then b=a. • Transitive property: If a=b and b=c, then a=c. • Substitution property: If a=b, then a can be substituted for b in any equation or expression.

40. Example • Solve 55z – 3(9z + 12) = -64 and write a reason for each step.

41. Properties of Equality

42. Review • Let p be “a shape is a triangle” and let q be “it has an acute angle”. • Write the contrapositive of p q. • Write the inverse of p q.

43. Review • How is the product 4 · 6 related to 52? • How is the product 5 · 7 related to 62? • Make a conjecture about how the product of two positive inters n and n + 2 is related to the square of the integer between them. • Write a convincing argument to justify your conjecture.

44. Pg. 91 # 8 – 20 • Pg. 92 # 26 – 42 • Pg. 99 # 15 – 23, 32

45. 2.5 Proving Statements and Segments Day 3 Part 1 CA Standard 2.0, 4.0, 16.0

46. Warmup • Identify the property of equality. If m<1 = m<2, then m<2 = m<1.

47. Properties of Segment Congruence • Segment congruence is reflexive, symmetric, and transitive. • Reflexive: For any segment AB, . • Symmetric: If , then . • Transitive: If , and , then

48. Symmetric Property of Segment Congruence • Given: PQ XY • Prove: XY PQ Statements Reasons 1.PQ XY 1. Given 2.PQ = XY 2. Definition of congruent segments 3.XY = PQ 3. Symmetric property of equality 4.XY PQ 4. Definition of congruent segments P X Q Y

49. Example

50. Another example • Given: LK = 5, JK = 5, JK JL. • Prove: LK JL. Statements Reasons 1._________ 1. Given 2. _________ 2. Given 3. LK = JK 3. Transitive p. o. e 4. LK JK 4. ________________ 5. JK JL 5. Given 6. ________ 6. Transitive p o e LK = 5 JK = 5 Definition of congruent segments LK ≈ JL