Chapter 2: Reasoning and Proof
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1 Name: Chapter 2: Reasoning and Proof Guided Notes Geometry Fall Semester
2 2.1 Use Inductive Reasoning CH. 2 Guided Notes, page 2 Term Definition Example conjecture An unproven statement that is based on observations. inductive reasoning The process of finding a pattern for specific cases and writing a conjecture for the general case. counterexample A specific example for which the conjecture is false. Examples: 1. Describe how to sketch the fourth figure in the pattern. Then sketch the fourth figure. 2. Describe the pattern in the numbers -1, -4, -16, -64,..... Then write the next three numbers in the pattern.
3 CH. 2 Guided Notes, page 3 3. Given five noncollinear points, make a conjecture about the number of different ways to connect the points. Number of Points Picture Numbers such as 1, 3, and 5 are called consecutive odd numbers. Make and test a conjecture about the sum of three consecutive odd numbers. Step one: Find a pattern using groups of small numbers. Step two: Make a conjecture. Step three: Test your conjecture. 5. A student makes the following conjecture about the difference of two numbers. Find a counterexample to disprove the student s conjecture. Student s conjecture: The difference of any two numbers is always smaller than the larger number. (Hint: To find a counterexample you need to find an example that is opposite what the student is saying. Prove him/her wrong using an example.)
4 CH. 2 Guided Notes, page 4 6. This scatter plot shows the average salary of players in the National Football League (NFL) since Make a conjecture based on the graph.
5 2.2 Analyze Conditional Statements CH. 2 Guided Notes, page 5 Term Definition Example conditional statements (p q) A logical statement that has two parts, a hypothesis and a conclusion. if-then form hypothesis (p) A form of a conditional statement in which the if part contains the hypothesis and the then part contains the conclusion. The if part of a conditional statement. conclusion (q) The then part of a conditional statement. negation (~p) The opposite of the original statement. converse (q p) Formed by switching the hypothesis and conclusion. inverse (~p ~q) Formed by negating both the hypothesis and conclusion. contrapositive (~q ~p) Formed by writing the converse and then negating both the hypothesis and conclusion.
6 equivalent statements Two statements that are both true or both false. CH. 2 Guided Notes, page 6 perpendicular lines Two lines that intersect to form right angles. biconditional statements (p! q) A statement that contains the phrase if and only if. Examples: 1. Rewrite the conditional statement in If-then form. Statement: All vertebrates have a backbone. If-then form: If, then. 2. Write the If-then form, the converse, the inverse, and the contrapositive of the conditional statement... Olympians are athletes. Decide whether each statmenet is true or false. If-then form: Converse: Inverse:
7 CH. 2 Guided Notes, page 7 Contrapositive: 3. Decide whether each statement about the diagram is true. Explain your answer using the definitions you have learned. a. AC! BD b.!aed and!bec are a linear pair. 4. Write the definition of parallel lines as a biconditional. Definition: If two lines lie in the same plane and do not intersect, then they are parallel. Converse: Biconditional:
8 2.3 Apply Deductive Reasoning CH. 2 Guided Notes, page 8 Term Definition Example deductive reasoning Using facts, definitions, accepted properties, and the laws of logic to form an argument. Law of Detachment If p q is a true conditional and p is true, then q is true. Also called a direct argument. Law of Syllogism If p q and q r are true conditionals, then p r is also true. Also called the chain rule. Examples: 1. Use the Law of Detachment to make a valid conclusion statement. a). If two angles have the same measure, then they are congruent. You are given that m!a = m!b. Hypothesis:. Conclusion:. Valid conclusion:.
9 CH. 2 Guided Notes, page 9 b). Jesse goes to the gym every weekday. Today is Monday. Write as if-then statement:. Hypothesis:. Conclusion:. Valid conclusion:. 2. If possible, use the Law of Syllogism to write the conditional statement that follows from the pair of true statements. a). If Ron eats lunch today, then he will eat a sandwich. If Ron eats a sandwich, then he will drink a glass of milk. Identify parts of first conditional statement: Hypothesis:. Conclusion:. Identify parts of second conditional statement: Hypothesis:. Conclusion:. New conditional statement using Law of Syllogism:. b). If x 2 > 36, then x 2 > 30. If x > 6, then x 2 > st statement: Hyp.! Concl.! 2 nd statement: Hyp.! Concl.! New conditional statement using the Law of Syllogism:.
10 CH. 2 Guided Notes, page 10 c). If a triangle is equilateral, then all of its sides are congruent. If a triangle is equilateral, then all angles in the interior of the triangle are congruent. 1 st statement: Hyp! Concl.! 2 nd statement: Hyp.! Concl.! New conditional statement using the Law of Syllogism:. 3. Inductive or Deductive Reasoning? Remember... Inductive Reasoning is based on observation and pattern. We don t always know whether our conjecture is true. Deductive Reasoning is based on fact. Tell whether the statement is the result of inductive or deductive reasoning. a). The runner s average speed decreases as the time spent running increases. b). The runner s average speed is slower when running for 40 minutes than when running for 10 minutes.
11 CH. 2 Guided Notes, page 11 Chapter 2 Extension: Symbolic Notation and Truth Tables Term Definition Example truth value truth table
12 2.4 Use Postulates and Diagrams CH. 2 Guided Notes, page 12 Term Definition Example Postulate 1 Postulate 2 Postulate 3 Postulate 4 Ruler Postulate Segment Addition Postulate Protractor Postulate Angle Addition Postulate 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. 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. Postulate 11 If two planes intersect, then their intersection is a line. line perpendicular to a plane A line is! to a plane if and only if the line intersects the plane at a point that is! to every line on the plane.
13 CH. 2 Guided Notes, page Reason Using Properties from Algebra Algebraic Properties of Equality Let a, b, and c be real numbers. 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 Substitution Property Distributive Property If a = b, then a can be substituted for b in any equation or expression. a(b + c) = ab + ac, where a, b, and c are real numbers. Properties of Equality Property Real Numbers Segments Angles Reflexive For any real number For any segment AB, For any angle A, a, a = a. AB = BA. m A = m! A!. Symmetric For any real numbers For any segments AB and For any angles A and B, if a and b, if a = b, CD, if AB = CD, then m A = m! B!, then then b = a. CD = AB. m B = m! A!. Transitive For any real numbers For any segments AB, CD, For any angles A, B, and C, a, b, and c, if a = b and EF, if AB = CD and if m A = m! B! and and b = c, then CD = EF, then AB = EF. m B = m! C!, then a = c. m A = m! C!.
14 Examples: CH. 2 Guided Notes, page Solve the following equation and write reasons for each step. STEP REASON 1. 2x + 3 = 9! x Solve, using the Distributive Property. Write reasons for each step. 1.!4(6x + 2) = A motorist travels 5 miles per hour slower than the speed limit (s) for 3.5 hours. The distance traveled (d) can be determined by the formula d = 3.5(s! 5). Solve for s. Write reasons for each step. 1. d = 3.5(s! 5)
15 4. Use properties of equality to show that CF = AD. Take your given statements from the diagram. Equation CH. 2 Guided Notes, page 15 Reason
16 CH. 2 Guided Notes, page Prove Statements about Segments and Angles Term Definition Example proof two-column proof theorem Theorem 2.1 Congruence of Segments Segment congruence is reflexive, symmetric, and transitive. Reflexive For any segment AB, AB! AB. Symmetric If AB! CD, then CD! AB. Transitive If AB! CD and CD! EF, then AB! EF. Theorem 2.2 Congruence of Angles Angle congruence is reflexive, symmetric, and transitive. Reflexive For any angle A, m A " m! A!. Symmetric If m! A " m! B, then m B " m! A!. Transitive If m! A " m! B and m! B " m! C, then m A " m! C!. Midpoint Definition in Proofs: Congruent Angles and Segments definition in Proofs: Angle Bisector Definition in Proofs:
17 Examples: CH. 2 Guided Notes, page WRITE A TWO-COLUM PROOF Use the diagram to prove that m!1 = m!4. Given: m!2 = m!3, m!axd = m!axc Prove: m!1 = m!4 Statements Reasons 1. m!axc = m!axd m!axd = m!1 + m! m!axc = m!3 + m! m!1 + m!2 = m!3 + m! m!2 = m! m!1 + m!3 = m!3 + m! m!1 = m! Name the property that is illustrated by the following statement. If!5 "!3, then!3 "!5. 3. If you know that BD bisects!abc, prove that m!abc is two times m!1. Given: BD bisects!abc Prove: m!abc = 2 m!1 Statements Reasons 1. BD bisects!abc ! bisector!! " s 3. 3.! " s!= " s 4. m!1 + m!2 = m!abc m!1 + m! = m!abc 5. Subst. P.O.E Simplifiy
18 CH. 2 Guided Notes, page 18 4 Complete the following two-column proof. Given: R is the midpoint of AM and MB = AR. Prove: M is the midpoint of RB. Statements Reasons 1. R is the midpoint of AM. 1. MB = AR 2. AR! RM AR = RM Trans. P.O.E. 5. MB! RM M is the midpoint of RB 6.
19 2.7 Prove Angle Pair Relationships CH. 2 Guided Notes, page 19 Term Definition Example Theorem 2.3 Right Angles Congruence Theorem All right angles are congruent. linear pair Postulate 12 Linear Pair Postulate If two angles form a linear pair, then they are supplementary. vertical angles Theorem 2.6 Vertical Angles Congruence Theorem Vertical angles are congruent. Examples: 1. Given: JK!KL, ML!KL Prove:!K "!L Statements Reasons 1. JK!KL, ML!KL ! " rt.# s 3.!K "!L 3.
20 CH. 2 Guided Notes, page Given:!4 is a right angle Prove: m!2 = 90 Statements Reasons 1.!4 is a right angle m!4 = !2 "! m!2 = m! m!2 = Use the diagram to decide if the statement is true or false. a) If m!1 = 47, then m!2 = 43. b) If m!1 = 47, then m!3 = 47. c) m!1 + m!3 = m!2 + m!4 d) m!1 + m!4 = m!2 + m!3
21 CH. 2 Guided Notes, page Find the value of the variables and the measure of each angle in the diagram.
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