Chapter 2 Deductive Reasoning Learn deductive logic Do your first 2- column proof New Theorems and Postulates **PUT YOUR LAWYER HAT ON!!

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 Conditional: is a two part statement with an actual or implied if-then. If p, then q. p ---> q 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

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 exactly one plane contains them The whole thing: If two lines intersect, then exactly one plane contains them. (Theorem 1 – 3) All theorems, postulates, and definitions are conditional statements!!

The Converse A conditional with the hypothesis and conclusion reversed. If q, then p. q ---> p hypothesis conclusion If it is daytime, then the sun is shining. Original: If the sun is shining, then it is daytime. **BE AWARE, THE CONVERSE IS NOT ALWAYS TRUE!!

Using the same hypothesis as the statement, but coming to a different conclusion. *Like a lawyer providing an alibi for his client… The Counterexample

If p, then q TRUE FALSE **You need only a single counterexample to prove a statement false.

The Counterexample If x > 5, then x = 6. If x = 5, then 4x = 20 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.

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

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. The Biconditional Statement: If m  1 = 90 , then  1 is a right angle. Converse: If  1 is a right angle, then m  1 = 90 . m  1 = 90  if and only if  1 is a right angle.  1 is a right angle if and only if m  1 = 90 .

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

Circle the hypothesis and underline the conclusion 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

Circle the hypothesis and underline the conclusion K is the midpoint of JL only if JK = KL

Circle the hypothesis and underline the conclusion K is the midpoint of JL only if JK = KL

Provide a counterexample to show that each statement is false. If a number is divisible by 4, then it is divisible by 6.

Circle the hypothesis and underline the conclusion I’ll dive if you dive

Circle the hypothesis and underline the conclusion I’ll dive if you dive

Provide a counterexample to show that each statement is false. If x 2 = 49, then x = 7.

Circle the hypothesis and underline the conclusion r + n = s + n if r = s

Circle the hypothesis and underline the conclusion r + n = s + n if r = s

Provide a counterexample to show that each statement is false. If AB  BC, then B is the midpoint of AC.

WARM UP Is the original statement T or F? Then write the converse… if false, provide a counter example. If 3 points are in line, then they are colinnear. If 3 points are colinnear, then they are in line. If I live in Los Angeles, then I live in CA. If I live in CA, then I live in Los Angleles. –False, you could live in San Diego

2.2 Properties from Algebra Objectives Do your first proof Use the properties of algebra and the properties of congruence in proofs

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

Properties of Equality Addition Property Add prop of = Subtraction Property Subtr. Prop of = Multiplication Property Multp. Prop of = Division Property Div. Prop of = Substitution Property Substitution Numbers, variables, lengths, and angle measures WHAT I DO TO ONE SIDE OF THE EQUATION, I MUST DO …

Reflexive Property x = x. A number equals itself. Reflexive Prop. Transitive Property if x = y and y = z, then x = z. Two numbers equal to the same number are equal to each other. Transitive Pop. Properties of Equality Reflexive Property AB ≅ AB A segment (or angle) is congruent to itself Reflex. Prop Transitive Property 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. Prop Properties of Congruence

Rules of Thumb…. Measurements are = –(prop. of equality) Figures are  –(Prop. of congruencey)

Whiteboards Page 40 –#’s 1 – 10

Your First Proof Given: 3x x = 22 Prove: x = x x = Given 2. -5x + 7 = Substitution 3. -5x = Subtraction Prop. = 4. x = Division Prop. = (specifics) (general rules) STATEMENTS REASONS

Day 2 - How to write a proof Walk-Thru of examples on page 38 and 39

Given : WX = YZ Y is the midpoint of XZ Prove: WX = XY W ZY X Your Second Proof ** Before we actually do this as a proof, lets make a verbal argument about why this is true.**

StatementsReasons 1. Y is the midpoint of XZ 1. Given 2. XY = YZ2. Def of midpoint 3. WX = YZ3. Given 4. WX = XY4. Substitution

Your third Proof Given: AB = CD Prove: AC = BD 1. AB = CD 1. Given 2. BC = BC 2. Reflexive prop. 3. AB + BC = BC + CD 3. Addition Prop. = 4. AB + BC = AC 4. Segment Addition Post. BC + CD = BD 5. AC = BD 5. Substitution STATEMENTS REASONS A B C D

Whiteboards Pg. 40 #11

Warm-up Page. 40 #12 Discuss with class

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

PB & J Sandwich WARM – UP 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

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. 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.

Important Notes Does the order matter? –How do I get from CV to Glendale High? Don’t leave out steps  Don’t Assume

Being a lawyer… When making your case, you might reference laws, statutes, and/or previous cases in order to make your argument… YOU BETTER MAKE SURE YOU ARE REFERECING THE CORRECT ONES OR THE JUDGE WILL KICK YOU OUT OF THE COURTROOM!!

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?

The Angle Bisector Theorem If BX is the bisector of  ABC, then m  ABX = ½ m  ABC m  XBC = ½ m  ABC A B X C

Writing Proofs of Theorems Extract the given and the prove from the conditional statement –“If” is our given information –“then” is the information that must be proved Mark the diagram, create a game plan, and tackle the proof!

Reasons Used in Proofs (pg. 45) Given Information Definitions (bi-conditional) Postulates Properties of equality and congruence Theorems

Whiteboards Pg. 45 # 1-9

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 4 A B C D

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) StatementsReasons Etc…

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

StatementsReasons 1.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 W ZY X

StatementsReasons 1. WX = YZ Y is the midpoint of XZ 1. Given 2. XY = YZ2. Def of midpoint 3. WX = XY3. 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)

WARM – UP Answer true or false. If false, write a one sentence explanation. 1.The converse of a true statement is sometimes false. 2.Only one counterexample is needed to disprove a statement. 3.Properties of equality cannot be used in geometric proofs. 4.Postulates are deduced from theorems. 5.Every angle has only one bisector.

Draw diagram on bottom of page 51 to reference during lesson ( add a line to make vertical angles)

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.. 1.We are always referring to a pair of angles (2 angles).. No more no less 2.Angles DO NOT have to be adjacent 3.**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 X T A B C  ABC is the complement of  SXT  SXT is the complement of  ABC

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

Complimentary & Supplementary angles Rules that apply to either type.. 1.We are always referring to a pair of angles (2 angles).. No more no less 2.Angles DO NOT have to be adjacent 3.**Do not get confused with the angle addition postulate 4.In proofs, you must first prove two L ’s add up to 90 or 180 before saying they are comp or suppl. NEED TO BE EXPLICT!!

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

A- Sometimes B – Always C - Never Two right angles are ____________ complementary.

Vertical Angles Two angles formed on the opposite sides of the intersection of two lines

Vertical Angles Two angles formed on the opposite sides of the intersection of two lines

Vertical Angles Two angles formed on the opposite sides of the intersection of two lines The only thing the definition does is identify what vertical angles are… NEVER USE THE DEFINITION IN A PROOF!!!

Theorem Vertical angles are congruent (The definition of Vert. angles does not tell us anything about congruency… this theorem proves that they are.) **THIS THEOREM WILL BE USED IN YOUR PROOFS OVER AND OVER

A- Sometimes B – Always C - Never Vertical angles ___________ have a common supplement.

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

If  1 and  2 are vertical angles, m  1 = 2x+18, and m  2 = 3x+4, then x = ???? 14

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

Whiteboard Pg.51 #9 Pg. 52 #10 – 19 (diagram)

Warm – Up Student will complete #33 from page 54 on front board

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

Perpendicular Lines (  ) If two intersecting lines form right angles, then they are perpendicular. If l  m, then the angles are right. l m If the angles are right, then l  m. What can you conclude about the rest of the angles in the diagram and why?

Two lines that form one right angle form four right angles The definition applies to intersecting rays and segments The definition can be used in two ways (bi- conditional) –PG. 56 Perpendicular Lines (  )

White Boards Page 57 –#1, 4, 5

White Boards Line AB  Line CD. G D F B C E A

Construction 4 Given a segment, construct the perpendicular bisector of the segment. Given: Construct: Steps:

Explanation of Theorem 2-4 If two lines are perpendicular, then they form congruent, adjacent angles. l m 1 2 If l  m, then  1   2. PARTNERS: Using the definition of perp. lines…what can you conclude about angles 1 and 2? Why?

Explanation of Theorem 2-5 If two lines intersect to form congruent, adjacent angles, then the lines are perpendicular. l m 1 2 If  1   2, then l  m. PARTNERS: Using what you know about the angles …. What can be concluded about the lines? Why

Theorem. m l 1 2 If l  m, then  1 and  2 are compl. What can you deduce about angle 1 and 2 based on the diagram?

PARTNERS Answer questions 6-10 on page 57 THE KEY IS TO UNDERSTAND THE EVIDENCE AND THE CONCLUSION IT BRINGS!! THE SPECFIC STATEMET FOLLOWS THE LOGIC OF THE GENERAL RULE #6 – Def. of perp. lines #7 – 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 5 Given a point on a line, construct the perpendicular to the line through the point. Given: Construct: Steps:

Construction 6 Given a point outside a line, construct the perpendicular to the line through the point. Given: Construct: Steps:

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

Partner Work in notes Page 62 (split paper in half – 1 prob on L one on right) –Copy down problem #1 –Copy down problem #5 Have the students come to a conclusion for each Explain to someone on the street that knows nothing about math about how you came to your conclusions…use an example as if complementary and supplementary represented a $$$$ amount. #8 – Right out an argument to explain the proof

Remember Magical Proof Steps

Demo Complimentary and supplementary Theorems Money Example

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  

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  

Partner – Candy Q’s Pg. 62 # 1-6

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

` StatementsReasons 1. m L 1 = m L 41. Given 2. m L 2 +m L 1 = angle addition postulate 3. L 2 + L 4 = 1803.Substituion 4. L 4 is supp. to L 2 4. Def of suppl L’s

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

StatementsReasons 1. L 2 and L 3 are supp.1. Given 2. m L 2 +m L 1 = angle addition postulate 3. L 2 is supp. to L 13. Def of supp. angles 4. m L 1 = m L 34. If two angles are supp. to the same angle, then the two angles are congruent

Test Review 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