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Chapter 2 Reasoning and Proof Chapter 2: Reasoning and Proof 1 Lesson 2 – 1 Conditional Statements Objectives 1 2 • Recognize conditional statements • To write converses of conditional statements Chapter 2: Reasoning and Proof 2 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 3 Lesson 2 – 1 Conditional Statements Identify the hypothesis and the conclusion: If two lines are parallel, then the lines are coplanar. Hypothesis: Conclusion: Chapter 2: Reasoning and Proof 4 Lesson 2 – 1 Conditional Statements 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 5 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 6 Lesson 2 – 1 Conditional Statements Find a counterexample to show that this conditional is false: If x2 ≥ 0, then x ≥ 0. Chapter 2: Reasoning and Proof 7 Lesson 2 – 1 Conditional Statements 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 8 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 pq Converse: q p Chapter 2: Reasoning and Proof If q, then p 9 Lesson 2 – 1 Conditional Statements 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 10 Lesson 2 – 1 Conditional Statements Write the converse of the conditional: If x = 9, then x + 3 = 12. Chapter 2: Reasoning and Proof 11 Lesson 2 – 1 Conditional Statements Write the converse of the conditional, and determine the truth value of each: If a2 = 25, a = 5. Chapter 2: Reasoning and Proof 12 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 13 Lesson 2 – 1 Conditional Statements Homework Pages 83-85 1, 2 -14 even, 28 -32 even, 40, 48-52 even Chapter 2: Reasoning and Proof 14 Lesson 2 – 2 Biconditionals and Definitions Objectives 1 2 •To write biconditionals •To recognize good definitions Chapter 2: Reasoning and Proof 15 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 16 Lesson 2 – 2 Biconditionals and Definitions 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 17 Lesson 2 – 2 Biconditionals and Definitions 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 18 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 19 Lesson 2 – 2 Biconditionals and Definitions 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 20 Lesson 2 – 2 Biconditionals and Definitions Is the following statement a good definition? Explain. An apple is a fruit that contains seeds. Chapter 2: Reasoning and Proof 21 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 22 Lesson 2 – 2 Biconditionals and Definitions Homework Page 90 2-26 even Chapter 2: Reasoning and Proof 23 Lesson 2 – 3 Deductive Reasoning Objectives 1 2 •To use the Law of Detachment •To use the Law of Syllogism Chapter 2: Reasoning and Proof 24 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 25 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 26 Lesson 2 – 3 Deductive Reasoning 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 27 Lesson 2 – 3 Deductive Reasoning 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 28 Lesson 2 – 3 Deductive Reasoning 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 29 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 30 Lesson 2 – 3 Deductive Reasoning 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 31 Lesson 2 – 3 Deductive Reasoning 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 32 Lesson 2 – 3 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 33 Lesson 2 – 3 Deductive Reasoning Homework Pages 96 – 97 1 – 21 Chapter 2: Reasoning and Proof 34 Lesson 2 – 4 Reasoning in Algebra Objectives 1 •To connect reasoning in algebra and geometry Chapter 2: Reasoning and Proof 35 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 a b If a = b and c ≠ 0, then c c Chapter 2: Reasoning and Proof 36 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 37 Lesson 2 – 4 Reasoning in Algebra Key Concepts The Distributive Property a(b + c) = ab + ac Chapter 2: Reasoning and Proof 38 Lesson 2 – 4 Reasoning in Algebra 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 39 Lesson 2 – 4 Reasoning in Algebra 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 40 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 41 Lesson 2 – 4 Reasoning in Algebra 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 42 Lesson 2 – 4 Reasoning in Algebra (continued) c. If ∠P ≅ ∠Q, ∠Q ≅ ∠R, and ∠R ≅ ∠S, then ∠P ≅ ∠S Chapter 2: Reasoning and Proof 43 Lesson 2 – 4 Reasoning in Algebra Lesson Quiz Name the justification for each statement. 1. ab = ab 2. If mABC + 40 = 85, then mABC = 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 mAOB + mBOC = mAOC. 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 44 Lesson 2 – 4 Reasoning in Algebra Homework Pages 105 – 107 2 – 22 even, 28, 31 Chapter 2: Reasoning and Proof 45 Lesson 2 – 5 Proving Angles Congruent Objectives 1 •To prove and apply theorems about angles Chapter 2: Reasoning and Proof 46 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 47 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, m1 + m 3 = 180 and m 2 + m 3 = 180. 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 48 Lesson 2 – 5 Proving Angles Congruent Find the value of x. Chapter 2: Reasoning and Proof 49 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, m1 + 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 50 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, m1 + 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 51 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 52 Lesson 2 – 5 Proving Angles Congruent 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 ● ● ● ● 53 Lesson 2 – 5 Proving Angles Congruent Lesson Quiz Use the diagram and mABS = 3x + 6 and mRBC = 5x – 20 for Exercises 1–4. 1. Find x. 2. Find mABS. 3. Find mSBC. 4. Without using the Vertical Angle Theorem, what theorem can you use to prove that ABR SBC? Chapter 2: Reasoning and Proof 54 Lesson 2 – 5 Proving Angles Congruent Homework Pages 112 – 114 1 – 7, 8 – 18 even, 21, 23 – 28 Chapter 2: Reasoning and Proof 55