Chapter 2 Deductive Reasoning

Chapter 2 Deductive Reasoning
Learn deductive logic Do your first 2-column proof New Theorems and Postulates **PUT YOUR LAWYER HAT ON!! In algebra, the emphasis is on solving problems and getting right answers. In Geometry, the emphasis is on how to solve problems, but not the answer. In a proof, we already know the answer. The purpose of a proof is to show why the answer is true.

2.1 If – Then Statements Objectives
Recognize the hypothesis and conclusion of an if-then statement State the converse of an if-then statement Use a counterexample Understand if and only if The if-then statement forms the basis for a syllogistic argument, which is the logical form used in a proof.

The If-Then Statement Conditional:
is a two part statement with an actual or implied if-then. If p, then q. p ---> q Logicians replace the phrases in an argument with letters to emphasize the form of the argument, rather than its meaning. Whether an argument is valid or invalid has nothing to do with the meaning, and with the words stripped away, this is easier to see. Once words are inserted into a conditional, we start to attach meaning to them, and try to evaluate the validity of the argument based on the meaning of the words: is it true or false? Truth or falsity have nothing to do with validity, so arguments are better left in symbols when validity is being examined. You use if-then statements to form a chain of logical deductions hypothesis conclusion If the sun is shining, then it is daytime.

Circle the hypothesis and underline the conclusion
If a = b, then a + c = b + c 4

Hidden If-Thens A conditional may not contain either if or then!
All theorems, postulates, and definitions are conditional statements!! Hidden If-Thens A conditional may not contain either if or then! Two intersecting lines are contained in exactly one plane. Which is the hypothesis? Which is the conclusion? two lines intersect All theorems, postulates, properties and definitions are conditionals. So is the previous sentence. exactly one plane contains them The whole thing: If two lines intersect, then exactly one plane contains them. (Theorem 1 – 3)

The Converse A conditional with the hypothesis and conclusion
reversed. Original: If the sun is shining, then it is daytime. If q, then p. q ---> p hypothesis conclusion Many students confuse the converse with the conditional, and assume that the logic runs both ways. A careful analysis will keep them from making that mistake. A statement and its converse can say different things. Some true statements have false converses. If it is daytime, then the sun is shining. **BE AWARE, THE CONVERSE IS NOT ALWAYS TRUE!!

The Counterexample An if –then statement is false if an example can be found for which the hypothesis is true and the conclusion is false. The example is called the Counterexample. *Like a lawyer providing an alibi for his client… Definitions, Theorems and postulates have no counterexample. Otherwise they would not be true. A conditional with only a single counterexample is false. To be true, it must always be true, with no exceptions.

The Counterexample If p, then q FALSE TRUE
Definitions, Theorems, and postulates have no counterexample. Otherwise they would not be true. A conditional with only a single counterexample is false. To be true, it must always be true, with no exceptions. **You need only a single counterexample to prove a statement false.

The Counterexample If x > 5, then x = 6.
x could be equal to 5.5 or 7 etc… always true, no counterexample **Definitions, Theorems and postulates have no counterexample. Otherwise they would not be true. To be true, it must always be true, with no exceptions. Theorems and postulates have no counterexample. Otherwise they would not be true. A conditional with only a single counterexample is false. To be true, it must always be true, with no exceptions.

Other Forms What do you notice? If p, then q p implies q p only if q
q if p Conditional statements are not always written with the “if” clause first. All of these conditionals mean the same thing. What do you notice? 10

The Biconditional If a conditional and its converse are the same (both true) then it is a biconditional and can use the “if and only if” language. Statement: If m1 = 90, then 1 is a right angle. Converse: If 1 is a right angle, then m1 = 90. Use the definition of a right angle to demonstrate a biconditional. Stress that although the conditional and its’ converse must have the same truth, they need not be true. m1 = 90 if and only if 1 is a right angle. 1 is a right angle if and only if m1 = 90 . 11

White Board Practice Circle the hypothesis and underline the conclusion VW = XY implies VW  XY

VW = XY implies VW  XY

Write the converse of each statement
If I play football, then I am an athlete If I am an athlete, then I play football If 2x = 4, then x = 2 If x = 2, then 2x = 4

Provide a counterexample to show that each statement is false.
If a line lies in a vertical plane, then the line is vertical

K is the midpoint of JL only if JK = KL

If a number is divisible by 4, then it is divisible by 6. 18

n > 8 only if n is greater than 7

I’ll dive if you dive

If x2 = 49, then x = 7. 23

r + n = s + n if r = s

If AB  BC, then B is the midpoint of AC. 26

2.2 Properties from Algebra
Objectives Do your first proof Use the properties of algebra and the properties of congruence in proofs Proofs baffle most geometry students at the start. One of the best ways to ensure that they make the connection between the given and the prove is to link it to something that they know: algebra.

Properties from Algebra
see properties on page 37 Read the first paragraph This lesson reviews the algebraic properties of equality that will be used to write proofs and solve problems. We treat the properties of Algebra like postulates Meaning we assume them to be true Go through the properties of equality and congruency and talk about what each one means. Indicate that they already know how to solve problems with these properties: they do so every time they solve an algebraic equation. Now they are going to use these properties like postulates and use them to prove why an equation has the solution that it does.

Properties of Equality
Numbers, variables, lengths, and angle measures WHAT I DO TO ONE SIDE OF THE EQUATION, I MUST DO … Addition Property if x = y, then x + z = y + z. Add prop of = Subtraction Property if x = y, then x – z = y – z. Subtr. Prop of = Multiplication Property if x = y, then xz = yz. Multp. Prop of = Division if x = y, and z ≠ 0, then x/z = y/z. Div. Prop of Come up with examples based on algebraic problems that you have done in the past When you write these in your proofs, you will write “________ property of =“ (which explains that it is a property of equality)

Order of equality does not matter. Symmetric Prop.
Substitution Property if x = y, then either x or y may be substituted for the other in any equation. Reflexive x = x. A number equals itself. Reflexive Prop. Symmetric if x = y, then y = x. Order of equality does not matter. Symmetric Prop. Transitive if x = y and y = z, then x = z. Two numbers equal to the same number are equal to each other. Transitive Pop. When talking about substitution property explain how (5x – 2x) is the same as 3x Just write… “substitution” The last three properties highlighted in yellow lead to the three properties of congruence.

Your First Proof Given: 3x + 7 - 8x = 22 Prove: x = - 3
(specifics) (general rules) STATEMENTS REASONS 1. 3x x = Given x + 7 = Substitution x = Subtraction Prop. = 4. x = Division Prop. = Take this very slowly. Have them look at the structure and format of the proof using this example. Note that the answer is already provided: doing a proof has nothing to do with finding the answer, it is all about justifying the answer logically.

Properties of Congruence
Segments, angles and polygons Reflexive Property AB ≅ AB A segment (or angle) is congruent to itself Reflex. Prop Symmetric Property If AB ≅ CD, then CD ≅ AB Order of equality does not matter. Symm. Transitive If AB ≅ CD and CD ≅ EF, then AB ≅ EF Two segments (or angles) congruent to the same segment (or angle) are congruent to each other. Trans. The properties of congruence follow from the properties of equality Since these are properties of both equality and congruence, when we write them as reasons in proofs we just put “_________ property” (no equal sign or congruent sign)

Your Second Proof Given: AB = CD Prove: AC = BD STATEMENTS REASONS
1. AB = CD Given 2. BC = BC Reflexive prop. 3. AB + BC = BC + CD Addition Prop. = 4. AB + BC = AC Segment Addition Post. BC + CD = BD 5. AC = BD Substitution Also take your time with this one. Show them to mark the given on the figure. Have them look at the figure and the given and put into words why this conclusion is true. If they can see it as true, they have already done the proof in their heads. To write the proof, they have to slow down their thought process and discover what the connections are between the given and the prove line. Have them write down all the postulates and properties that they have learned that might help with this proof (SAP, Addition Property of Equality, Substitution) and then help them to see how to place the statements and reasons so the whole is linked together from start (given) to finish (prove).

2.3 Proving Theorems Objectives
Use the Midpoint Theorem and the Bisector Theorem Know the kinds of reasons that can be used in proofs

YUMMY !

PB & J Sandwich How do I make one?
Pretend as if I have never made a PB & J sandwich. Not only have I never made one, I have never seen one or heard about a sandwich for that matter. Write out detailed instructions in full sentences I will collect this

First, open the bread package by untwisting the twist tie
First, open the bread package by untwisting the twist tie. Take out two slices of bread set one of these pieces aside. Set the other in front of you on a plate and remove the lid from the container with the peanut butter in it.

Take the knife, place it in the container of peanut butter, and with the knife, remove approximately a tablespoon of peanut butter. The amount is not terribly relevant, as long as it does not fall off the knife. Take the knife with the peanut butter on it and spread it on the slice of bread you have in front of you.

Repeat until the bread is reasonably covered on one side with peanut butter. At this point, you should wipe excess peanut butter on the inside rim of the peanut butter jar and set the knife on the counter.

Replace the lid on the peanut butter jar and set it aside
Replace the lid on the peanut butter jar and set it aside. Take the jar of jelly and repeat the process for peanut butter. As soon as you have finished this, take the slice of bread that you set aside earlier and place it on the slice with the peanut butter and jelly on it, so that the peanut butter and jelly is reasonably well contained within.

The Midpoint Theorem If M is the midpoint of AB, then
AM = ½ AB and MB = ½ AB This is the first theorem that can be proven with a 2-column proof. Show them how to extract the given and prove from the conditional statement that is the theorem. Mark the figure, take an inventory and do the proof. Distinguish between the Definition of a Midpoint and this theorem. They both talk about a midpoint, and are often confused. What is their difference? (one talks about congruent segments, while the other talks about something being half of something else) How would you know which one to use in a proof? (Does the proof feature fractions of one-half?) B A M

The Midpoint Theorem If M is the midpoint of AB, then
AM = ½ AB and MB = ½ AB How is the definition of a midpoint different from this theorem? One talks about congruent segments One talks about something being half of something else How do you know which one to use in a proof? This is the first theorem that can be proven with a 2-column proof. Show them how to extract the given and prove from the conditional statement that is the theorem. Mark the figure, take an inventory and do the proof. Distinguish between the Definition of a Midpoint and this theorem. They both talk about a midpoint, and are often confused. What is their difference? (one talks about congruent segments, while the other talks about something being half of something else) How would you know which one to use in a proof? (Does the proof feature fractions of one-half?)

Important Notes Does the order matter?
Don’t leave out steps  Don’t Assume

Given: M is the midpoint of AB
Prove: AM = ½ AB and MB = ½ AB B M A Statements (specifics) Reasons (general rules) 1. M is the midpoint of AB 1. Given 2. AM  MB or AM = MB 2. Definition of a midpoint 3. AM + MB = AB 3. Segment Addition Postulate 4. AM + AM = AB 4. Substitution Property Or 2 AM = AB 5. AM = ½ AB 5. Division Property of Equality 6. MB = ½ AB 6. Substitution

The Angle Bisector Theorem
If BX is the bisector of ABC, then m  ABX = ½ m  ABC m  XBC = ½ m  ABC A Ditto. Does this theorem sound familiar? Leave the other proof on the board and have them prove this one on their own. X B C

A Given: BX is the bisector of ABC Prove: m  ABX = ½ m  ABC m  XBC = ½ m  ABC X B C 1. BX is the bisector of  ABC; 1. Given 2. m  ABX = m  XBC 2. Definition of Angle Bisector or  ABX  m  XBC 3.m ABX + m  XBC = m ABC 3. Angle Addition Postulate 4.m ABX + m ABX = m ABC 4. Substitution 2 m  ABX = m  ABC 5. m  ABX = ½ m  ABC 5. Division Property of = 6. m  XBC = ½ m  ABC 6. Substitution Property

Reasons Used in Proofs (pg. 45)
Given Information Definitions (bi-conditional) Postulates Properties of equality and congruence Theorems They should begin organizing the statements they learn in class every day by making 3x5 note cards with the significant theorems, postulates, properties and definitions on them.

How to write a proof (The magical steps)
Use these steps every time you have to do a proof in class, for homework, on a test, etc.

Example 1 Given : m  1 = m  2; AD bisects  CAB; BD bisects  CBA
Prove: m  3 = m  4 D 2 1 3 4 B A 49

1. Copy down the problem. Write down the given and prove statements and draw the picture. Do this every single time, I don’t care that it is the same picture, or that the picture is in the book. Draw big pictures Use straight lines

2. Mark on the picture Read the given information and, if possible, make some kind of marking on the picture. Remember if the given information doesn’t exactly say something, then you must think of a valid reason why you can make the mark on the picture. Use different colors when you are marking on the picture.

3. Look at the picture This is where it is really important to know your postulates and theorems. Look for information that is FREE, but be careful not to Assume anything. Angle or Segment Addition Postulate Vertical angles Shared sides or angles Parallel line theorems

4. Brain. Do you have one?

I mean have you drawn a brain and are you writing down your thought process? Every single time you make any mark on the picture, you should have a specific reason why you can make this mark. If you can do this, then when you fill the brain the proof is practically done.

5. Finally look at what you are trying to prove
Ask yourself: “Does it make sense?” “why?” Write out a plan to help organize your thoughts Then try to work backwards and fill in any missing links in your brain. Think about how you can get that final statement.

6. Write the proof. (This should be the easy part) Statements Reasons
Etc… Etc…

Example 1 Given : m  1 = m  2; AD bisects  CAB; BD bisects  CBA
Prove: m  3 = m  4 D 2 1 3 4 B A

Statements Reasons m 1 = m  2; AD bisects  CAB; BD bisects  CBA 1. Given 2. m 1 = m  3; m 2 = m  4 2. Def of  bisector 3. m 3 = m  4 3. Substitution

Try it Given : WX = YZ Y is the midpoint of XZ Prove: WX = XY Y Z W X

Statements Reasons 1. WX = YZ Y is the midpoint of XZ 1. Given 2. XY = YZ 2. Def of midpoint 3. WX = XY 3. Substitution

QUIZ REVIEW Underline the hypothesis and conclusion in each statement
Write a converse of each statement and tell whether it is true or false Provide a counter example to show that the statement is false Be able to complete a proof Name the reasons used in a proof (there are 5)

2.4 Special Pairs of Angles
Objectives Apply the definitions of complimentary and supplementary angles State and apply the theorem about vertical angles

Complimentary & Supplementary angles
Rules that apply to either type.. We are always referring to a pair of angles (2 angles) .. No more no less Angles DO NOT have to be adjacent **Do not get confused with the angle addition postulate

Complimentary Angles Any two angles whose measures add up to 90.
If mABC + m SXT = 90, then  ABC and  SXT are complimentary. S A  ABC is the complement of  SXT  SXT is the complement of  ABC Complimentary angles do not have to be adjacent. This is the major difference between this definition and the AAP. To use the AAP, the angles have to be adjacent. X C B T

Supplementary Angles Any two angles whose measures sum to 180.
If mABC + m SXT = 180, then  ABC and  SXT are supplementary. S A Ditto.  ABC is the supplement of  SXT  SXT is the supplement of  ABC X C B T

Vertical Angles Two angles formed on the opposite sides of the intersection of two lines. 1 This is just the definition. It does not say anything useful, other than identify what vertical angles are. Are vertical angles congruent? Why? While the definition does not say they are congruent, there is a theorem that does… 4 2 3

Vertical Angles Two angles formed on the opposite sides of the intersection of two lines. 1 4 2 3 This is just the definition. It does not say anything useful, other than identify what vertical angles are. Are vertical angles congruent? Why? While the definition does not say they are congruent, there is a theorem that does… The only thing the definition does is identify what vertical angles are… NEVER USE THE DEFINITION IN A PROOF!!!

**THIS THEOREM WILL BE USED IN YOUR PROOFS OVER AND OVER
Vertical angles are congruent (The definition of Vert. angles does not tell us anything about congruency… this theorem proves that they are.) 1 Do a proof of this theorem, if there is time. 4 2 3

Remote Time

True or False m  A + m  B + m  C = 180, then ,  B, and  C are supplementary.

True or False Vertical angles have the same measure

True or False If  1 and  2 are vertical angles, m  1 = 2x+18, and m  2 = 3x+4, then x = 14.

A- Sometimes B – Always C - Never
Vertical angles ____________ have a common vertex.

Two right angles are ____________ complementary.

Right angles are ___________ vertical angles.

Angles A, B, and C are __________ complementary.

Vertical angles ___________ have a common supplement.

White Board Practice Find the measure of a complement and a supplement of  T. m  T = 40

White Board Practice Find the measure of a complement and a supplement of  T. m  T = 89

White Board Practice Find the measure of a complement and a supplement of  T. m  T = a

White Board Practice A supplement of an angle is three times as large as a complement of the angle. Find the measure of the angle. Let x = the measure of the angle. 180 – x : This is the supplement 90 – x : This is the complement 180 – x = 3 (90 – x) 180 – x = 270 – 3x 2x = 90 x = 45

2.5 Perpendicular Lines Objectives Recognize perpendicular lines
Objectives Recognize perpendicular lines Use the theorems about perpendicular lines

Perpendicular Lines ()
Two lines that intersect to form right angles. If l  m, then the angles are right. l Notice that the definition does not say anything about 90. It takes the definition of a right angle to do that. If the angles are right, then l  m. m

Perpendicular Lines ()
Two lines that form one right angle form four right angles You can conclude that two lines are perpendicular by definition, once you know that any of the angles they form is a right angle The definition applies to intersecting rays and segments The definition can be used in two ways (bi-conditional) PG. 56

Theorem If two lines are perpendicular, then they form congruent, adjacent angles. l If l  m, then 1   2. 1 2 m Do the proof of this theorem, if there is time. It is more difficult than it looks, but it shows the difference between the definition of perpendicular and the definition of a right angle, both of which are used in the proof. PARTNERS: Complete the proof on page 57 problem #1

Theorem If two lines intersect to form congruent, adjacent angles, then the lines are perpendicular. l If 1   2, then l  m. This is the converse to theorem Several theorem come in conversive pairs, where one is the converse of the other. Most students mistake on for the other, so spend some time talking about when they would use each, and the importance of direction in a conditional statement. 1 2 m

Partners THINK – PAIR – SHARE
Discuss the wording of Theorems 2 – 4 and 2 – 5. Look at the hypothesis and conclusion of each When would you use each in a proof?

Theorem If the exterior sides of two adjacent angles lie on perpendicular lines, then the angles are complimentary. l If l  m, then 1 and  2 are compl. 1 A picture says a thousand words, but this one only says 17 words. The only time you can use theorem 2-6 is with this figure. Notice also that it does not say the angles add up to 90, it just says they are complimentary. 2 m CAN ANYONE EXPLAIN?

PARTNERS Answer questions 6-10 on page 57 #6 – Def. of perp. lines
#8 – If 2 lines are perp., then they form cong. Adj. angles #9 – Def. of perp. Lines #10 – IF 2 lines form cong. Adj. angles, then the lines are perp.

Construction 4 Given a segment, construct the perpendicular bisector of the segment. Given: Construct: Steps: Do the construction for them on the board. The steps are in the textbook, and can be written down later.

Construction 5 Given a point on a line, construct the perpendicular to the line through the point. Given: Construct: Steps: Do the construction for them on the board. The steps are in the textbook, and can be written down later.

Construction 6 Given a point outside a line, construct the perpendicular to the line through the point. Given: Construct: Steps: Do the construction for them on the board. The steps are in the textbook, and can be written down later.

2.6 Planning a Proof Objectives
Discover the steps used to plan a proof

Remember Magical Proof Steps

Demo Complimentary and supplementary Theorems I need 4 Volunteers

Theorem If two angles are supplementary to congruent angles (or the same angle) then they are congruent. If 1 suppl  2 and  2 suppl  3, then  1   3. 1 Show several configurations of angles for this theorem. Notice how this theorem is like substitution, for supplementary. Do the proof of this theorem (or 2-8) 2 3

Theorem If two angles are complimentary to congruent angles (or to the same angle) then they are congruent. If 1 compl  2 and  2 compl  3, then  1   3. 1 2 3

Practice Given:  2 and  3 are supplementary Prove: m  1 = m  3 3 1
4

Statements Reasons 1. L2 and L3 are supp. 1. Given 2. mL2 +m L1 = 180 2. angle addition postulate 3. L2 is supp. to L1 3. Def of supp. angles 4. mL1 = mL3 4.If two angles are supp. to the same angle, then the two angles are congruent

Practice Given: m  1 = m  4 Prove:  4 is supplementary to  2 3 1 2

Statements Reasons 1. mL1 = m L4 1. Given 2. mL2 +m L1 = 180 2. angle addition postulate 3. L2 is supp. to L1 3. Def of supp. angles 4. L4 is supp. to L2 4.Substituion

Test Review Name the following
Underline the hypothesis and conclusion in each statement Write a converse of each statement and tell whether it is true or false Fill in the blanks of an algebraic proof and a geometric proof Name the following Complementary / supplementary angles Perpendicular lines or rays Vertical angles

Understand what you can deduce from a diagram that is marked
Right angles = 90 / Straight angles that = 180 Using vertical angles to find measures Setting up an algebraic problem of = angles in order to solve for a variable – then using the variable to solve the measure of other angles **SHOWING YOUR WORK IN THE ANSWER DOCUMENT WHEN SOLVING FOR A VARIABLE OR MEASUREMENT** Setting up and solving an equation involving a supplement and complement of an angle Complete an entire geometric proof

Chapter 2 Deductive Reasoning

Similar presentations

Presentation on theme: "Chapter 2 Deductive Reasoning"— Presentation transcript:

Similar presentations

About project

Feedback

Log in

Auth with social network:

Chapter 2 Deductive Reasoning

Similar presentations

Presentation on theme: "Chapter 2 Deductive Reasoning"— Presentation transcript:

Similar presentations

About project

Feedback