Chapter 2 Reasoning and Proof
Chapter Objectives Recognize conditional statements Compare bi-conditional statements and definitions Utilize deductive reasoning Apply certain properties of algebra to geometrical properties Write postulates about the basic components of geometry Derive Vertical Angles Theorem Prove Linear Pair Postulate Identify reflexive, symmetric and transitive
Conditional Statements Lesson 2.1 Conditional Statements
Lesson 2.1 Objectives Analyze conditional statements Write postulates about points, lines, and planes using conditional statements
Conditional Statements A conditional statement is any statement that is written, or can be written, in the if-then form. This is a logical statement that contains two parts Hypothesis Conclusion If today is Tuesday, then tomorrow is Wednesday.
Hypothesis The hypothesis of a conditional statement is the portion that has, or can be written, with the word if in front. When asked to identify the hypothesis, you do not include the word if. If today is Tuesday, then tomorrow is Wednesday.
Conclusion The conclusion of a conditional statement is the portion that has, or can be written with, the phrase then in front of it. Again, do not include the word then when asked to identify the conclusion. If today is Tuesday, then tomorrow is Wednesday.
Converse The converse of a conditional statement is formed by switching the hypothesis and conclusion. If today is Tuesday, then tomorrow is Wednesday. If tomorrow is Wednesday, then today is Tuesday
Negation The negation is the opposite of the original statement. Make the statement negative of what it was. Use phrases like Not, no, un, never, can’t, will not, nor, wouldn’t, etc. Today is Tuesday. Today is not Tuesday.
Inverse The inverse is found by negating the hypothesis and the conclusion. Notice the order remains the same! If today is Tuesday, then tomorrow is Wednesday. If today is not Tuesday, then tomorrow is not Wednesday.
Contrapositive The contrapositive is formed by switching the order and making both negative. If today is Tuesday, then tomorrow is Wednesday. If today is not Tuesday, then tomorrow is not Wednesday. If tomorrow is not Wednesday, then today is not Tuesday.
Point, Line, Plane Postulates: Postulate 5 Through any two points there exists exactly one line. Y O
Point, Line, Plane Postulates: Postulate 6 A line contains at least two points. Taking Postulate 5 and Postulate 6 together tells you that all you need is two points to make one line. H I
Point, Line, Plane Postulates: Postulate 7 If two lines intersect, then their intersection is exactly one point. B
Point, Line, Plane Postulates: Postulate 8 Through any three noncollinear points there exists exactly one plane. M R L
Point, Line, Plane Postulates: Postulate 9 A plane contains at least three noncollinear points. Take Postulate 8 with Postulate 9 and this says you only need three points to make a plane. M R L
Point, Line, Plane Postulates: Postulate 10 If two points lie in a plane, then the line containing them lies in the same plane. M E
Point, Line, Plane Postulates: Postulate 11 If two planes intersect, then their intersection is a line. Imagine that the walls of the classroom are different planes. Ask yourself where do they intersect? And what geometric figure do they form?
Homework 2.1 In Class Homework Due Tomorrow 1-8 10-50 ev, 51, 55, 56 p75-78 Homework 10-50 ev, 51, 55, 56 Due Tomorrow
Definitions and Biconditional Statements Lesson 2.2 Definitions and Biconditional Statements
Lesson 2.2 Objectives Recognize a definition Recognize a biconditional statement Verify definitions using biconditional statements
Perpendicular Lines Perpendicular lines intersect to form a right angle. When writing that lines are perpendicular, we place a special symbol between the line segments AB CD T
Definition The previous slide was an example of a definition. It can be read forwards or backwards and maintain truth.
Biconditional Statement A biconditional statement is a statement that is written, or can be written, with the phrase if and only if. If and only if can be written shorthand by iff. Writing a biconditional is equivalent to writing a conditional and its converse. All definitions are biconditional statements.
Finding Counterexamples To find a counterexample, use the following method Assume that the hypothesis is TRUE. Find any example that would make the conclusion FALSE. For a biconditional statement, you must prove that both the original conditional statement has no counterexamples and that its converse has no counterexamples. If either of them have a counterexample, then the whole thing is FALSE.
Example 1 If a+b is even, then both a and b must be even. Assume that the hypothesis is TRUE. So pick a number that is even (larger than 2) Find any example that would make the conclusion FALSE. Pick two numbers that are not even but add to equal the even number from above. Those two numbers you picked are your counterexample. If no counterexample can be found, then the statement is true.
Homework 2.2 In Class 3-12 p82-85 Homework 14-42 even Due Tomorrow
Lesson 2.3 Deductive Reasoning
Lesson 2.3 Objectives Use symbolic notation to represent conditional statements Identify the symbol for negation Utilize the Law of Detachment to form conclusions Utilize the Law of Syllogism to form conclusions
Symbolic Conditional Statements To represent the hypothesis symbolically, we use the letter p. We are applying algebra to logic by representing entire phrases using the letter p. To represent the conclusion, we use the letter q. To represent the phrase if…then, we use an arrow, . To represent the phrase if and only if, we use a two headed arrow, .
Example of Symbolic Representation If today is Tuesday, then tomorrow is Wednesday. p = Today is Tuesday q = Tomorrow is Wednesday Symbolic form p q We read it to say “If p then q.”
Negation Recall that negation makes the statement “negative.” That is done by inserting the words not, nor, or, neither, etc. The symbol is much like a negative sign but slightly altered… ~
Symbolic Variations Converse Inverse Contrapositive Biconditional q p Inverse ~p ~q Contrapositive ~q ~p Biconditional p q
Logical Argument Deductive reasoning uses facts, definitions, and accepted properties in a logical order to write a logical argument. So deductive reasoning either states laws and/or conditional statements that can be written in if…then form. There are two laws that govern deductive reasoning. If the logical argument follows one of those laws, then it is said to be valid, or true.
Law of Detachment If pq is a true conditional statement and p is true, then q is true. It should be stated to you that pq is true. Then it will describe that p happened. So you can assume that q is going to happen also. This law is best recognized when you are told that the hypothesis of the conditional statement happened.
Example 2 If you get a D- or above in Geometry, then you will get credit for the class. Your final grade is a D. Therefore… You will get credit for this class!
Law of Syllogism If pq and qr are true conditional statements, then pr is true. This is like combining two conditional statements into one conditional statement. The new conditional statement is found by taking the hypothesis of the first conditional and using the conclusion of the second. This law is best recognized when multiple conditional statements are given to you and they share alike phrases.
Example 3 If tomorrow is Wednesday, then the day after is Thursday. If the day after is Thursday, then there is a quiz on Thursday. Therefore… And this gets phrased using another conditional statement If tomorrow is Wednesday, then there is a quiz on Thursday.
Deductive v Inductive Reasoning Deductive reasoning uses facts, definitions, and accepted properties in a logical order to write a proof. This is often called a logical argument. Inductive reasoning uses patterns of a sample population to predict the behavior of the entire population This involves making conjectures based on observations of the sample population to describe the entire population.
Equivalent Statements If the conditional statement is true, then the contrapositive is also true. Therefore they are equivalent statements! Equivalent Statements Conditional Converse Inverse Contrapositive If p, then q If q, then p If ~p, then ~q If ~q, then ~p Written just as it shows in the problem. Switch the hypothesis with the conclusion. Take the original conditional statement and make both parts negative. Take the converse and make both parts negative. If the converse is true, then the inverse is also true. Therefore they are equivalent statements! Means “not”
Homework In Class 1-5 p91-94 Homework 8-48 even Due Tomorrow
Reasoning with Properties of Algebra Lesson 2.4 Reasoning with Properties of Algebra
Lesson 2.4 Objectives Use properties from algebra to create a proof Utilize properties of length and measure to justify segment and angle relationships
Algebraic Properties of Equality Property Definition Identification Abbreviation Addition If a=b, then a+c = b+c. Something is added to both sides of the equation. APOE Subtraction If a=b, then a-c = b-c. Something is subtracted from both sides of the equation. SPOE Multiplication If a=b, then ac = bc. Something is multiplied to both sides of the equation. MPOE Division If a=b and c≠0, then a/c = b/c. Something is being divided into both sides. DPOE Substitution If a=b, then a can be substituted for b in any expression. One object is used in place of another without any calculations being done. SUB Distributive a(b+c) = ab + ac A number outside of parentheses has been multiplied to all numbers inside. DIST
Reflexive, Symmetric and Transitive Properties Definition For any real number a, a = a If a=b, then b=a. If a=b and b=c, then a=c. How to Remember Reflexive is close to reflection, which is what you see when you look in a mirror. Symmetric starts with s, so that means to switch the order. Transitive is like transition, and when a and c equal the same thing, they must transition to equal each other. How to Use This will be used when two objects share something, such as sharing a common side of a triangle This is a step that allows you to change the order of objects so they fit where you need them. This is used most often in proofs, and can be often thought of as substitution.
Show Your Work This section is an introduction to proofs. To solve any algebra problem, you now need to show ALL steps. And with those steps you need to give a reason, or law, that allows you to make that step. Remember to list your first step by simply rewriting the problem. This is to signify how the problem started.
Example 4 Solve 9x+18=72 9x+18=72 Given 9x=54 SPOE x=6 DPOE -18 9 Short for “Information given to us.” 9x+18=72 Given -18 9x=54 SPOE 9 x=6 DPOE
Example 5: Using Segments In the diagram, AB=CD. Show that AC=BD. A B C D Think about changing AB into AC? And the same with CD into BD? AB=CD Given AB+BC=BC+CD APOE Segment Addition Postulate AC=AB+BC Segment Addition Postulate BD=BC+CD AC=BD Transitive POE
Example 6: Using Angles HW Problem #24, p100 In the diagram, m RPQ=m RPS, verify to show that m SPQ=2(m RPQ). mRPQ=m RPS Given Angle Addition Postulate m SPQ=m RPQ+m RPS S R Q P m SPQ=m RPQ+m RPQ SUB m SPQ=2(m RPQ) DIST
Example 7 Fill in the two-column proof with the appropriate reasons for each step APOE MPOE Symmetric POE
Homework 2.4 In Class Homework Due Tomorrow 1,4-8 10-32, 36-50 even p99-101 Homework 10-32, 36-50 even Due Tomorrow
Proving Statements about Segments Lesson 2.5 Proving Statements about Segments
Lesson 2.5 Objectives Write a two-column proof Justify statements about congruent segments
Theorem A theorem is a true statement that follows the truth of other statements. Theorems are derived from postulates, definitions, and other theorems. All theorems must be proved.
Two-Column Proof One method of proving a theorem is to use a two-column proof. A two-column proof has numbered statements and corresponding reasons placed in a logical order. That logical order is just steps to follow much like reading a cook book. The first step in a two-column proof should always be rewriting the information given to you in the problem. When you write your reason for this step, you say “Given”. The last step in a two-column proof is the exact statement that you are asked to show.
Example 8 Prove the Symmetric Property of Segment Congruence. GIVEN: Segment PQ is congruent to Segment XY PROVE: Segment XY is congruent to Segment PQ
Hints for Making Proofs Remember to always write down the first step as given information. Develop a mental plan of how you want to change the first statement to look like the last statement. Try to evaluate how you can make each step change from the previous by applying some rule. You must follow the postulates, definitions, and theorems that you already know. Number your steps so the statements and the reasons match up!
Example 9 Fill in the missing steps Transitive POE A C
Example 10 Fill in the missing steps 1 and 2 are a linear pair 1 and 2 are supplementary Definition of supplementary angles m1 = 180o - m2
Homework 2.5 In Class Homework Due Tomorrow 1,3-5,7,9 6-11,16,21,22 p105-107 Homework 6-11,16,21,22 Due Tomorrow
Proving Statements about Angles Lesson 2.6 Proving Statements about Angles
Lesson 2.6 Objectives Utilize the angle and segment congruence properties Prove properties about special angle pairs
Theorem 2.1: Properties of Segment Congruence Segment congruence is always Reflexive Segment AB is congruent to Segment AB. Symmetric If AB CD, then CD AB. Transitive If AB CD and CD EF, then AB EF.
Theorem 2.2: Properties of Angle Congruence Angle congruence is always Reflexive A A Symmetric If A B, then B A. Transititve If A B and B C, then A C.
Theorem 2.3: Right Angle Congruence Theorem All right angles are congruent. 1 2 GIVEN: 1 and 2 are right angles. PROVE: 1 2 1. 1 and 2 are right angles 1. Given 2. m1 = 90o, m2 = 90o 2. Definition of Right Angles 3. m1 = m2 3. Trans POE 4. 1 2 4. DEFCON
Theorem 2.4: Congruent Supplements Theorem If two angles are supplementary to the same angle, or congruent angles, then they are congruent. If m1 + 2 = 180o and m2 + m3 = 180o, then 1 3. 2 1 3
Theorem 2.5: Congruent Complements Theorem If two angles are complementary to the same angle, or to congruent angles, then they are congruent. If m4 + m5 = 90o and m5 + m6 =90o, then 4 6. 5 4 6
Postulate 12: Linear Pair Postulate The Linear Pair Postulate says if two angles form a linear pair, then they are supplementary. 1 + 2 = 180o 1 2
Theorem 2.6: Vertical Angles Theorem If two angles are vertical angles, then they are congruent. Vertical angles are angles formed by the intersection of two straight lines. 1 3 2 1 3 4 2 4
Example 11 Using the following figure, fill in the missing steps to the proof. Given 2 Definition of a linear pair 4 m1 + m2 = 180o m3 + m4 = 180o Congruent Supplements Theorem
Homework 2.6 In Class Homework Due Tomorrow 1,3-9,10,23 p112-116 Homework 10, 12-22, 27-28, 33-36 Due Tomorrow