Chapter 2 Reasoning and Proof

Patterns and Inductive Reasoning Lesson 2.1 Patterns and Inductive Reasoning

Introduction Geometry, like much of mathematics and science, developed when people began recognizing and describing patterns. 21 blocks 15 blocks 10 blocks 6 blocks ... 3 blocks 1 block

Example Describe a pattern in the sequence of numbers and predict the next number. Pattern: Each number is 4 times the previous number Pattern: Each number is 3 more than the previous number a) 1, 4, 16, 64, ... 256 b) 2, 5, 8, 11, ... 14

2 Types of Reasoning Inductive Reasoning – Looking at several previous examples and drawing a conjecture (educated guess) Deductive Reasoning – Uses facts, definitions, and other rules in order to draw a conclusion.

Using Inductive Reasoning Look for a Pattern – Look at several examples. Pictures and tables can be helpful. Make a Conjecture – Use examples to make a conjecture. Verify the Conjecture – Use deductive reasoning to the conjecture is true for all cases.

Example Complete the conjecture based on the pattern you observe in the specific cases. 1 + 1 = 2 7 + 11 = 18 1 + 3 = 4 13 + 19 = 32 Conjecture: The sum of any two odd numbers is ________.

Counterexample An example that proves a statement false. Ex. All four legged animals are dogs. counterexample: cats Ex. All red trucks are fire trucks. counterexample: Dodge Ram

Homework Page 85 6, 7,12, 13, 16, 20, 21-24, 25-28, 33-37, 38, 39

Conditional Statements Lesson 2.2 Conditional Statements

Conditional statements Conditional statement- a sentence that contains an “if” and a “then” phrase Ex. If you frown, then you are sad Ex. If an angle is right, then it is 90°

Hypothesis / Conclusion Hypothesis-the phrase that follows the word if Conclusion-the phrase that follows the word then Ex. If you frown, then you are sad Ex. If an angle is right, then it is 90° Ex. You are late if you arrive after the bell

How to rewrite in if-then form Identify the hypothesis and conclusion (look for an “if”). Rewrite in the form: If hypothesis then conclusion. Verify that the new statement has the same meaning as the initial statement.

Rewrite in if-then form Three points are collinear if they lie on the same line If they lie on the same line then three points are collinear If three points lie on the same line then they are collinear Cats make Ms. Wilson sneeze If cats then make Ms. Wilson sneeze If an animal is a cat then it makes Ms. Wilson sneeze

Different forms a conditional Inverse – negate both hypothesis and conclusion Converse – switch the places of the hypothesis and conclusion Contrapositive – negate both the hypothesis and conclusion and then switch their places (basically do inverse then converse)

(implies) Symbolic Notation p  q means “p implies q” p is the hypothesis and q is the conclusion It can be rewritten as “If p then q”

Symbolic Notation negation is a ~ given conditional: p  q inverse: ~ p  ~ q converse: q  p contrapostive: ~ q  ~ p

Negation Negation - change the “truth” Ex. The letter A is a vowel Done by: changing a word to its opposite inserting (or removing) the word NOT Ex. The letter A is a vowel Negation: The letter A is NOT a vowel The letter A is a consonant

True or False A conditional statement can either be true or false To show true - must show true for ALL cases To show false - must provide one time false called a counterexample

Writing the inverse, converse and contrapositive Given conditional: If an animal is a fish then it can swim First identify the hypothesis and conclusion Inverse: Converse: Contrapositive:

Is the conditional true or false? If false, provide a counterexample If two angles are a linear pair then they are supplementary angles If a number is odd, then it is divisible by 3

Homework 2-2 Practice (on website) Page 93 13-19

Biconditional Statements and Definitions Lesson 2.3 Biconditional Statements and Definitions

Biconditional statement A statement that contains “if and only if” (symbol ) The same as writing a statement and its converse all in one. To be “true” - statement and its converse must both be true

Definitions as conditionals Definitions can be written word first or description first A right angle has a measure of 90° If an angle is right then it has a measure of 90° If an angle has a measure of 90° then it is right All definitions can be rewritten as a true bi- conditional statement An angle is right if and only if it has a measure of 90°

Practice writing statements Perpendicular lines are two lines that intersect to form a right angle Write as a conditional statement: Write as its converse: Write as a bi-conditional statement:

More Practice X = 5 if and only if 2x = 10 What kind of statement is this? Re-write as a conditional. Write the converse of this conditional.

Homework Page 101 7-10, 13-15, 24, 25

Lesson 2-4 Deductive Reasoning

Laws of Logic Law of Ruling Out Possibilities Given: A or B is true and B is not true Our conclusion: A is true

Laws of Logic Law of Detachment Given: pq is true and p is true Our conclusion: q is true

Laws of Logic Law of Syllogism (Transitivity) Given: pq and qr Our conclusion: pr look for common phrase in one hypothesis and one conclusion

Example If the sun is out, then it is not cloudy The sun is out Our conclusion: It is not cloudy

Example If trees are around, then Ms. Wilson sneezes Ms. Wilson sneezed Our conclusion: NO conclusion (We can only make a conclusion if we know that trees are around)

Example If the sun is out, then we’ll go biking If we go biking, then we’ll be happy Our conclusion: If the sun is out, then we’ll be happy

Example If it is Friday, then there is a game tonight If it is Friday, then tomorrow is Saturday Conclusion: NO conclusion (The common phrase must be in one hypothesis and one conclusion)

Homework

Reasoning in Algebra and Geometry Lesson 2-5 Reasoning in Algebra and Geometry

Properties from Algebra 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 & c is not 0, then a / c = b / c

Properties from Algebra Distributive Property if a ( b + c ) then a * b + a * c Substitution Property if a = b, then a can be substituted for b in any equation or expression

Properties from Algebra Reflexive Property for any real number, a = a Symmetric Property if a = b, then b = a Transitive Property if a = b and b = c, then a = c

Example Multiplication Property State the property being demonstrated with each statement If x + 10 = 15 then x = 5 Subtraction Property If AB = 4cm then 5AB = 20cm Multiplication Property

Example Transitive Property MN Use the property to complete the statement Transitive Property If RB = MN and ____ = WY then RB = WY MN

Example Given Addition Property Subtraction Property Division Property Solve the equation and write a reason for each step 3x - 17 = x + 5 3x = x + 22 2x = 22 x = 11 Given Addition Property Subtraction Property Division Property

Example Solve the equation and write a reason for each step 55x – 3(9x + 12) = -64 55x – 27x – 36 = -64 28x - 36 = -64 28x = -28 x = -1 Given Distributive Property Combine Like Terms Addition Property Division Property

Homework Page 117 6-20

Proving Angles Congruent Lesson 2-6 Proving Angles Congruent

Proofs & Theorems No need to write this down Proof - statements and reasons that show the logical order of an argument Theorem – a statement that can be proven true

Steps of a Proof Draw the picture Make a “T” chart Statements on left / Reasons on right Input the given information Mark the drawing Make statements and reasons that lead to the “prove” statement

Special Types of Angles review from Chapter 1 Right – has a measure of 90° Complementary - 2 angles whose sum is 90° Supplementary - 2 angles whose sum is 180°

Vertical Angle Theorem Linear Pair Postulate If two angles form a linear pair, then they are supplementary Vertical Angle Theorem Vertical angles are congruent

Example Solve for x 2x + 18 = 6x - 190 Why? Vertical s are =

Complete the proof Statements Reason 2 1 3 Given Given: m1 = m2 2 and 3 are supplementary Prove: 1 and 3 are supplementary Statements Reason m1 = m2 m 2 + m 3 = 180° m 1 + m 3 = 180° 1 and 3 are supplementary 2 1 3 Given Def of supplementary s Substitution Property

Statements Reasons 6 8 5 7 Given: 6  7 Prove: 5  8 6  7 5  6 5  7 7  8 5  8 5 7 Given Vertical  Theorem Transitive Property

Homework

Law of Detacment -What does this mean? Given: One if-then sentence The “if” part is repeated Conclusion: The “then” part

Law of Syllogism - What does this mean? Given: Two if-then sentence A common phrase is once in the “if” part and once in the “then” part Conclusion: The non-common parts are put together into one if-then sentence