1. Chapter 2
Reasoning and Proof
Chapter 2: Reasoning and Proof

2. Conditional Statements
Lesson 2 – 1
Conditional Statements
Objectives 1
Recognize conditional statements 2
To write converses of conditional statements
Chapter 2: Reasoning and Proof

3. Conditional Statements
Lesson 2 – 1
Conditional Statements
Key Concepts
A conditional statement is _______________________.
Every conditional statement has two parts.
The part following the “If” is the ____________.
The part following the “then” is the __________.
Chapter 2: Reasoning and Proof

4. Conditional Statements
Lesson 2 – 1
Conditional Statements
Example 1
Identify the hypothesis and the conclusion:
If two lines are parallel, then the lines are coplanar.
Hypothesis:
Conclusion:
Chapter 2: Reasoning and Proof

5. Conditional Statements
Lesson 2 – 1
Conditional Statements
Example 2
Write the statement as a conditional:
An acute angle measures less than 90º.
The subject of the sentence is “An acute angle.”
The hypothesis is “An angle is acute.” The first part of the conditional is “If an angle is acute.”
The verb and object of the sentence are “measures less than 90°.”
The conclusion is “It measures less than 90°.” The second part of the conditional is “then it measures less than 90°.”
“If an angle is acute, then it measures less than 90°.”
Chapter 2: Reasoning and Proof

6. Conditional Statements
Lesson 2 – 1
Conditional Statements
Key Concepts
A _________________ is a case in which the hypothesis is true and the conclusion is false.
To show that a conditional is false, you need to find only one counterexample.
Chapter 2: Reasoning and Proof

7. Conditional Statements
Lesson 2 – 1
Conditional Statements
Example 3
Find a counterexample to show that this conditional is false: If x2 ≥ 0, then x ≥ 0.
Chapter 2: Reasoning and Proof

8. Conditional Statements
Lesson 2 – 1
Conditional Statements
Example 4
Use the Venn diagram below. What does it mean to be inside the large circle but outside the small circle?
Chapter 2: Reasoning and Proof

9. Conditional Statements
Lesson 2 – 1
Conditional Statements
Key Concepts
In the converse of a conditional statement the hypothesis and conclusion are switched.
Conditional: If p, then q
Converse: If q, then p
Chapter 2: Reasoning and Proof

10. Conditional Statements
Lesson 2 – 1
Conditional Statements
Example 5
The Mad Hatter states: “You might just as well say that ‘I see what I eat’ is the same thing as ‘I eat what I see’!” Provide a counterexample to show that one of the Mad Hatter’s statements is false.
Chapter 2: Reasoning and Proof

11. Conditional Statements
Lesson 2 – 1
Conditional Statements
Example 6
Write the converse of the conditional:
If x = 9, then x + 3 = 12.
Chapter 2: Reasoning and Proof

12. Conditional Statements
Lesson 2 – 1
Conditional Statements
Example 7
Write the converse of the conditional, and determine the truth value of each:
If a2 = 25, a = 5.
Chapter 2: Reasoning and Proof

13. Conditional Statements
Lesson 2 – 1
Conditional Statements
Lesson Quiz
Use the following conditional for Exercises 1–3.
If a circle’s radius is 2 m, then its diameter is 4 m.
1. Identify the hypothesis and conclusion.
2. Write the converse.
If a circle’s diameter is 4 m, then its radius is 2 m.
3. Determine the truth value of the conditional and its converse.
Both are true.
Show that each conditional is false by finding a counterexample.
4. If lines do not intersect, then they are parallel.
skew lines
5. All numbers containing the digit 0 are divisible by 10.
Sample: 105
Chapter 2: Reasoning and Proof

14. Conditional Statements
Lesson 2 – 1
Conditional Statements
Homework
Pages 83-85
1, even, even, 40, even
Chapter 2: Reasoning and Proof

15. Biconditionals and Definitions
Lesson 2 – 2
Biconditionals and Definitions
Objectives 1
To write biconditionals 2
To recognize good definitions
Chapter 2: Reasoning and Proof

16. Biconditionals and Definitions
Lesson 2 – 2
Biconditionals and Definitions
Key Concepts
If a conditional and its converse are both true, the statement is said to be ________________.
Biconditional statements are often stated in the form “…if and only if …”
IFF – short for if and only if
 - symbol for if and only if
An angle is a right angle if and only if it measures 90°.
Chapter 2: Reasoning and Proof

17. Biconditionals and Definitions
Lesson 2 – 2
Biconditionals and Definitions
Example 1
Consider this true conditional statement. Write its converse. If the converse is also true, combine the statements as a biconditional.
Chapter 2: Reasoning and Proof

18. Biconditionals and Definitions
Lesson 2 – 2
Biconditionals and Definitions
Example 2
Write the two statements that form this biconditional.
Biconditional: Lines are skew if and only if they are noncoplanar.
Conditional:
Converse:
Chapter 2: Reasoning and Proof

19. Biconditionals and Definitions
Lesson 2 – 2
Biconditionals and Definitions
Key Concepts
The Reversibility Test
The reverse (converse) of a definition must be true. If the reverse of a statement is false, then the statement is not a good definition.
A good definition is reversible. That means that you can write a good definition as a true biconditional.
Chapter 2: Reasoning and Proof

20. Biconditionals and Definitions
Lesson 2 – 2
Biconditionals and Definitions
Example 3
Show that this definition of triangle is reversible. Then write it as a true biconditional.
Definition: A triangle is a polygon with exactly three sides.
Chapter 2: Reasoning and Proof

21. Biconditionals and Definitions
Lesson 2 – 2
Biconditionals and Definitions
Example 4
Is the following statement a good definition? Explain.
An apple is a fruit that contains seeds.
Chapter 2: Reasoning and Proof

22. Biconditionals and Definitions
Lesson 2 – 2
Biconditionals and Definitions
Lesson Quiz
1. Write the converse of the statement.
If it rains, then the car gets wet.
2. Write the statement above and its converse as a biconditional.
3. Write the two conditional statements that make up the biconditional.
An angle is a straight angle if and only if it measures 180°.
Is each statement a good definition? If not, find a counterexample.
4. The midpoint of a line segment is the point that divides the segment into two congruent segments.
5. A line segment is a part of a line.
Chapter 2: Reasoning and Proof

23. Biconditionals and Definitions
Lesson 2 – 2
Biconditionals and Definitions
Homework
Page 90
2-26 even
Chapter 2: Reasoning and Proof

24. Deductive Reasoning Objectives Lesson 2 – 31
To use the Law of Detachment 2
To use the Law of Syllogism
Chapter 2: Reasoning and Proof

25. Deductive Reasoning Key Concepts
Lesson 2 – 3
Deductive Reasoning
Key Concepts
Deductive Reasoning (or logical reasoning) is
If the given statements are true, deductive
reasoning produces a true conclusion.
Chapter 2: Reasoning and Proof

26. Deductive Reasoning Key Concepts Law of Detachment
Lesson 2 – 3
Deductive Reasoning
Key Concepts
Law of Detachment
If a conditional is true and its hypothesis is true, then its conclusion is true.
In symbolic form:
If p  q is a true statement and p is true, then q is true.
Chapter 2: Reasoning and Proof

27. Lesson 2 – 3
Deductive Reasoning
Example 1
A gardener knows that if it rains, the garden will be watered. It is raining. What conclusion can he make?
Chapter 2: Reasoning and Proof

28. Deductive Reasoning For the given statements, what can you conclude?
Lesson 2 – 3
Deductive Reasoning
Example 2
For the given statements, what can you conclude?
Given: If an angle acute, then its measure is less than 90°. A is acute.
Chapter 2: Reasoning and Proof

29. Lesson 2 – 3
Deductive Reasoning
Example 3
Does the following argument illustrate the Law of Detachment?
Given: If you make a field goal in basketball, you score two points. Jenna scored two points in basketball.
Chapter 2: Reasoning and Proof

30. Deductive Reasoning Key Concepts Law of Syllogism
Lesson 2 – 3
Deductive Reasoning
Key Concepts
Law of Syllogism
If p  q and q  r are true statements, then p  r is a true statement.
Chapter 2: Reasoning and Proof

31. Lesson 2 – 3
Deductive Reasoning
Example 4
Use the Law of Syllogism to draw a conclusion from the following true statements:
If a quadrilateral is a square, then it contains four right angles.
If a quadrilateral contains four right angles, then it is a rectangle.
Chapter 2: Reasoning and Proof

32. Lesson 2 – 3
Deductive Reasoning
Example 5
Use the Laws of Detachment and Syllogism to draw a possible conclusion.
If the circus is in town, then there are tents at the fairground.
If there are tents at the fairground, then Paul is working as a night watchman. The circus is in town.
Chapter 2: Reasoning and Proof

33. Deductive Reasoning Lesson Quiz Use the three statements below.
A. If games are canceled, then Maria reads a book.
B. If it snows, then games are canceled.
C. It is snowing.
1. Using only statements A and B, what can you conclude?
2. Using only statements B and C, what can you conclude?
3. Using statements A, B, and C, what can you conclude?
4. Suppose both statement B and “games are canceled” are true. Can you conclude that statement C is true? Explain.
Chapter 2: Reasoning and Proof

34. Deductive Reasoning Pages 96 – 97 1 – 21 Homework Lesson 2 – 3
Chapter 2: Reasoning and Proof

35. Reasoning in Algebra Objectives Lesson 2 – 41
To connect reasoning in algebra and geometry
Chapter 2: Reasoning and Proof

36. Reasoning in Algebra Key Concepts Properties of Equality
Lesson 2 – 4
Reasoning in Algebra
Key Concepts
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 a · c = b · c.
Division Property
If a = b and c ≠ 0, then
Chapter 2: Reasoning and Proof

37. Reasoning in Algebra Key Concepts Properties of Equality continued
Lesson 2 – 4
Reasoning in Algebra
Key Concepts
Properties of Equality continued
Reflexive Property
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 b can replace a in any expression.
Chapter 2: Reasoning and Proof

38. Reasoning in Algebra Key Concepts The Distributive Property
Lesson 2 – 4
Reasoning in Algebra
Key Concepts
The Distributive Property
a(b + c) = ab + ac
Chapter 2: Reasoning and Proof

39. Reasoning in Algebra Justify each step used to solve
Lesson 2 – 4
Reasoning in Algebra
Example 1
Justify each step used to solve
5x – 12 = 32 + x for x.
Given: 5x – 12 = 32 + x
1. 5x = 44 + x
2. 4x = 44
3. x = 11
Chapter 2: Reasoning and Proof

40. Lesson 2 – 4
Reasoning in Algebra
Example 2
Suppose that points A, B, and C are collinear with point B between points A and C.
Solve for x if AC = 21, BC = 15 – x, and
AB = 4 + 2x. Justify each step.
AB + BC = AC
(4 + 2x) + (15 – x) = 21
19 + x = 21
x = 2
Chapter 2: Reasoning and Proof

41. Reasoning in Algebra Key Concepts Properties of Congruence
Lesson 2 – 4
Reasoning in Algebra
Key Concepts
Properties of Congruence
Reflexive Property
AB ≅ AB ∠A≅∠A
Symmetric Property
If AB ≅ CD , then CD ≅ AB
If ∠A≅∠B, then ∠B≅∠A.
Transitive Property
If AB ≅ CD and CD ≅ EF , then AB ≅ EF
If ∠A≅∠B and ∠B≅∠C, then ∠A≅∠C.
Chapter 2: Reasoning and Proof

42. Reasoning in Algebra Name the property that justifies each statement.
Lesson 2 – 4
Reasoning in Algebra
Example 3
Name the property that justifies each statement.
a. If x = y and y + 4 = 3x, then x + 4 = 3x.
b. If x + 4 = 3x, then 4 = 2x.
Chapter 2: Reasoning and Proof

43. Reasoning in Algebra (continued)
Lesson 2 – 4
Reasoning in Algebra
Example 3
(continued)
c. If ∠P ≅ ∠Q, ∠Q ≅ ∠R, and ∠R ≅ ∠S, then ∠P ≅ ∠S
Chapter 2: Reasoning and Proof

44. Reasoning in Algebra Lesson Quiz
Name the justification for each statement.
1. ab = ab
2. If mABC + 40 = 85, then mABC = 45.
3. If k = m and k + w = 12, then m + w = 12.
4. If B is a point in the interior of AOC, then mAOB + mBOC = mAOC.
5. Fill in the missing information.
Given: AC = 36
a. AB + BC = AC i. ?
b. 3x + 2x + 1 = 36 ii. ?  
c. ?   iii. ?  
d. 5x = 35 iv. ?   
e. x = ?   v. ?  
Chapter 2: Reasoning and Proof

45. Reasoning in Algebra Pages 105 – 107 2 – 22 even, 28, 31 Homework
Lesson 2 – 4
Reasoning in Algebra
Homework
Pages 105 – 107
2 – 22 even, 28, 31
Chapter 2: Reasoning and Proof

46. Proving Angles Congruent
Lesson 2 – 5
Proving Angles Congruent
Objectives 1
To prove and apply theorems about angles
Chapter 2: Reasoning and Proof

47. Proving Angles Congruent
Lesson 2 – 5
Proving Angles Congruent
Key Concepts
A __________ is a convincing argument that uses deductive reasoning.
A statement that you prove true is a ____________.
A paragraph proof is written as sentences in a paragraph.
“Given”: lists what you know from the
hypothesis of the theorem
“Prove”: the conclusion of the theorem
Diagram: records the given information visually
Chapter 2: Reasoning and Proof

48. Proving Angles Congruent
Lesson 2 – 5
Proving Angles Congruent
Key Concepts
Theorem: Vertical angles are congruent.
Given: 1 and 2 are vertical angles
Prove: 1  2
Proof: By the Angle Addition Postulate,
m1 + m 3 = 180 and m 2 + m 3 = By substitution, m 1 + m 3 = m 2 + m 3. Subtract m 3 from each side. You get
m 1 = m 2, or  1   2.
Chapter 2: Reasoning and Proof

49. Proving Angles Congruent
Lesson 2 – 5
Proving Angles Congruent
Example 1
Find the value of x.
Chapter 2: Reasoning and Proof

50. Proving Angles Congruent
Lesson 2 – 5
Proving Angles Congruent
Key Concepts
Theorem: If two angles are supplements of the same angle, then the two angles are congruent.
Given: 1 and 2 are supplementary
3 and 2 are supplementary
Prove: 1  3
Proof: By the definition of supplementary angles,
m1 + m 2 = 180 and m 3 + m 2 = 180.
By substitution, m 1 + m 2 = m 3 + m 2.
Subtract m 2 from each side. You get m 1 = m 3, or  1   3.
Chapter 2: Reasoning and Proof

51. Proving Angles Congruent
Lesson 2 – 5
Proving Angles Congruent
Key Concepts
Theorem: If two angles are supplements of congruent angles, then the two angles are congruent.
Given: 1 and 2 are supplementary
3 and 4 are supplementary
2  4
Prove: 1  3
Proof: By the definition of supplementary angles,
m1 + m 2 = 180 and m 3 + m 4 = 180.
By substitution, m 1 + m 2 = m 3 + m 4.
Since 2  4, by the definition of congruence m 2 = m 4.
By substitution m 1 + m 4 = m 3 + m 4.
Subtract m 4 from each side. You get m 1 = m 3, or
 1   3.
Chapter 2: Reasoning and Proof

52. Proving Angles Congruent
Lesson 2 – 5
Proving Angles Congruent
Key Concepts
Theorem: If two angles are complements of the same angle (or of congruent angles), then the two angles are congruent.
Theorem: All right angles are congruent.
Theorem: If two angles are congruent and supplementary, then each is a right angle.
Chapter 2: Reasoning and Proof

53. Proving Angles Congruent
Lesson 2 – 5
Proving Angles Congruent
Example 2
Write a paragraph proof using the given, what you are to prove, and the diagram.
Given: WX = YZ ●
Prove: WY = XZ
Chapter 2: Reasoning and Proof

54. Proving Angles Congruent
Lesson 2 – 5
Proving Angles Congruent
Lesson Quiz
Use the diagram and mABS = 3x + 6 and
mRBC = 5x – 20 for Exercises 1–4.
1. Find x.
2. Find mABS.
3. Find mSBC.
4. Without using the Vertical Angle Theorem, what theorem can you use to prove that ABR  SBC?
Chapter 2: Reasoning and Proof

55. Proving Angles Congruent
Lesson 2 – 5
Proving Angles Congruent
Homework
Pages 112 – 114
1 – 7, 8 – 18 even, 21, 23 – 28
Chapter 2: Reasoning and Proof