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

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Conditional Statements Chapter 2: Reasoning and Proof2 Lesson 2 1 Objectives 1 Recognize conditional statements 2 To write converses of conditional statements

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Conditional Statements Chapter 2: Reasoning and Proof3 Lesson 2 1 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 __________.

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Conditional Statements Chapter 2: Reasoning and Proof4 Lesson 2 1 Identify the hypothesis and the conclusion: If two lines are parallel, then the lines are coplanar. Conclusion: Hypothesis:

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Conditional Statements Chapter 2: Reasoning and Proof5 Lesson 2 1 Write the statement as a conditional: An acute angle measures less than 90. If an angle is acute, then it measures less than 90. 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.

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Conditional Statements Chapter 2: Reasoning and Proof6 Lesson 2 1 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.

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Conditional Statements Chapter 2: Reasoning and Proof7 Lesson 2 1 Find a counterexample to show that this conditional is false: If x 2 0, then x 0.

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Conditional Statements Chapter 2: Reasoning and Proof8 Lesson 2 1 Use the Venn diagram below. What does it mean to be inside the large circle but outside the small circle?

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Conditional Statements Chapter 2: Reasoning and Proof9 Lesson 2 1 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

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Conditional Statements Chapter 2: Reasoning and Proof10 Lesson 2 1 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 Hatters statements is false.

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Conditional Statements Chapter 2: Reasoning and Proof11 Lesson 2 1 Write the converse of the conditional: If x = 9, then x + 3 = 12.

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Conditional Statements Chapter 2: Reasoning and Proof12 Lesson 2 1 Write the converse of the conditional, and determine the truth value of each: If a 2 = 25, a = 5.

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Conditional Statements Chapter 2: Reasoning and Proof13 Lesson 2 1 Lesson Quiz Use the following conditional for Exercises 13. If a circles radius is 2 m, then its diameter is 4 m. 1.Identify the hypothesis and conclusion. 2.Write the converse. If a circles 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

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Conditional Statements Chapter 2: Reasoning and Proof14 Lesson 2 1 Homework Pages 83-85 1, 2 -14 even, 28 -32 even, 40, 48-52 even

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Biconditionals and Definitions Chapter 2: Reasoning and Proof15 Lesson 2 2 Objectives 1 To write biconditionals 2 To recognize good definitions

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Biconditionals and Definitions Chapter 2: Reasoning and Proof16 Lesson 2 2 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.

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Biconditionals and Definitions Chapter 2: Reasoning and Proof17 Lesson 2 2 Consider this true conditional statement. Write its converse. If the converse is also true, combine the statements as a biconditional.

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Biconditionals and Definitions Chapter 2: Reasoning and Proof18 Lesson 2 2 Write the two statements that form this biconditional. Conditional: Converse: Biconditional: Lines are skew if and only if they are noncoplanar.

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Biconditionals and Definitions Chapter 2: Reasoning and Proof19 Lesson 2 2 Key Concepts A good definition is reversible. That means that you can write a good definition as a true biconditional. 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.

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Biconditionals and Definitions Chapter 2: Reasoning and Proof20 Lesson 2 2 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.

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Biconditionals and Definitions Chapter 2: Reasoning and Proof21 Lesson 2 2 Is the following statement a good definition? Explain. An apple is a fruit that contains seeds.

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Biconditionals and Definitions Chapter 2: Reasoning and Proof22 Lesson 2 2 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.

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Biconditionals and Definitions Chapter 2: Reasoning and Proof23 Lesson 2 2 Homework Page 90 2-26 even

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Deductive Reasoning Chapter 2: Reasoning and Proof24 Lesson 2 3 Objectives 1 To use the Law of Detachment 2 To use the Law of Syllogism

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Deductive Reasoning Chapter 2: Reasoning and Proof25 Lesson 2 3 Key Concepts Deductive Reasoning (or logical reasoning) is If the given statements are true, deductive reasoning produces a true conclusion.

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Deductive Reasoning Chapter 2: Reasoning and Proof26 Lesson 2 3 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.

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Deductive Reasoning Chapter 2: Reasoning and Proof27 Lesson 2 3 A gardener knows that if it rains, the garden will be watered. It is raining. What conclusion can he make?

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Deductive Reasoning Chapter 2: Reasoning and Proof28 Lesson 2 3 For the given statements, what can you conclude? Given: If an angle acute, then its measure is less than 90. A is acute.

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Deductive Reasoning Chapter 2: Reasoning and Proof29 Lesson 2 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.

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Deductive Reasoning Chapter 2: Reasoning and Proof30 Lesson 2 3 Key Concepts Law of Syllogism If p q and q r are true statements, then p r is a true statement.

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Deductive Reasoning Chapter 2: Reasoning and Proof31 Lesson 2 3 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.

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Deductive Reasoning Chapter 2: Reasoning and Proof32 Lesson 2 3 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.

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Deductive Reasoning Chapter 2: Reasoning and Proof33 Lesson 2 3 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.

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Deductive Reasoning Chapter 2: Reasoning and Proof34 Lesson 2 3 Homework Pages 96 97 1 21

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Reasoning in Algebra Chapter 2: Reasoning and Proof35 Lesson 2 4 Objectives 1 To connect reasoning in algebra and geometry

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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 Reasoning in Algebra Chapter 2: Reasoning and Proof36 Lesson 2 4 Key Concepts Properties of Equality

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Reasoning in Algebra Chapter 2: Reasoning and Proof37 Lesson 2 4 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.

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The Distributive Property a(b + c) = ab + ac Reasoning in Algebra Chapter 2: Reasoning and Proof38 Lesson 2 4 Key Concepts

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Reasoning in Algebra Chapter 2: Reasoning and Proof39 Lesson 2 4 Justify each step used to solve 5x 12 = 32 + x for x. 1.5x = 44 + x 2.4x = 44 3.x = 11 Given: 5x 12 = 32 + x

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Reasoning in Algebra Chapter 2: Reasoning and Proof40 Lesson 2 4 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=2x=2

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Reasoning in Algebra Chapter 2: Reasoning and Proof41 Lesson 2 4 Key Concepts Properties of Congruence

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Reasoning in Algebra Chapter 2: Reasoning and Proof42 Lesson 2 4 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.

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Reasoning in Algebra Chapter 2: Reasoning and Proof43 Lesson 2 4 (continued) c.If P Q, Q R, and R S, then P S

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Lesson Quiz Reasoning in Algebra Chapter 2: Reasoning and Proof44 Lesson 2 4 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 = ACi. ? b.3x + 2x + 1 = 36ii. ? c. ? iii. ? d.5x = 35iv. ? e.x = ? v. ?

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Reasoning in Algebra Chapter 2: Reasoning and Proof45 Lesson 2 4 Homework Pages 105 107 2 22 even, 28, 31

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Proving Angles Congruent Chapter 2: Reasoning and Proof46 Lesson 2 5 Objectives 1 To prove and apply theorems about angles

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Proving Angles Congruent Chapter 2: Reasoning and Proof47 Lesson 2 5 Key Concepts A statement that you prove true is a ____________. A paragraph proof is written as sentences in a paragraph. A __________ is a convincing argument that uses deductive reasoning. Given: lists what you know from the hypothesis of the theorem Prove: the conclusion of the theorem Diagram: records the given information visually

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Proving Angles Congruent Chapter 2: Reasoning and Proof48 Lesson 2 5 Key Concepts 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 = 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. Theorem: Vertical angles are congruent.

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Proving Angles Congruent Chapter 2: Reasoning and Proof49 Lesson 2 5 Find the value of x.

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Proving Angles Congruent Chapter 2: Reasoning and Proof50 Lesson 2 5 Key Concepts 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. Theorem: If two angles are supplements of the same angle, then the two angles are congruent.

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Proving Angles Congruent Chapter 2: Reasoning and Proof51 Lesson 2 5 Key Concepts 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. Theorem: If two angles are supplements of congruent angles, then the two angles are congruent.

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Proving Angles Congruent Chapter 2: Reasoning and Proof52 Lesson 2 5 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.

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Proving Angles Congruent Chapter 2: Reasoning and Proof53 Lesson 2 5 Write a paragraph proof using the given, what you are to prove, and the diagram. Given: WX = YZ Prove: WY = XZ

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Proving Angles Congruent Chapter 2: Reasoning and Proof54 Lesson 2 5 Lesson Quiz Use the diagram and m ABS = 3x + 6 and m RBC = 5x 20 for Exercises 14. 1. Find x. 2. Find m ABS. 4. Without using the Vertical Angle Theorem, what theorem can you use to prove that ABR SBC? 3. Find m SBC.

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Proving Angles Congruent Chapter 2: Reasoning and Proof55 Lesson 2 5 Homework Pages 112 114 1 7, 8 18 even, 21, 23 28