Chapter 2 Deductive Reasoning Learn deductive logic Do your first 2- column proof New Theorems and Postulates

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. hypothesis conclusion If the sun is shining, then it is daytime.

Hidden If-Thens A conditional may not contain either if or then! All of my students love Geometry. Which is the hypothesis? Which is the conclusion? You are my student you love Geometry If you are my student, then you love Geometry.

The Converse A conditional with the hypothesis and conclusion reversed. If q, then p. hypothesis conclusion If it is daytime, then the sun is shining. Original: If the sun is shining, then it is daytime.

The Counterexample If p, then q TRUE FALSE

The only way a conditional can be false is if the hypothesis is true and the conclusion is false. This is called a counterexample. The Counterexample

This is HARD !

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

Group Practice Provide a counterexample to show that each statement is false. If you live in California, then you live in La Crescenta.

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

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

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

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

Other Forms If p, then q p implies q p only if q q if p What do you notice?

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

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

Circle the hypothesis and underline the conclusion n > 8 only if n is greater than 7

Circle the hypothesis and underline the conclusion n > 8 only if n is greater than 7

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

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

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

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

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

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

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 If m  1 = 90 , then  1 is a right angle. If  1 is a right angle, then m  1 = 90 . m  1 = 90  if and only if  1 is a right angle. m  1 = 90  iff  1 is a right angle.

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

Properties of Equality Addition Property if x = y, then x + z = y + z. Subtraction Property if x = y, then x – z = y – z. Multiplication Property if x = y, then xz = yz. Division Property if x = y, and z ≠ 0, then x/z = y/z.

Substitution Property if x = y, then either x or y may be substituted for the other in any equation. Reflexive Property x = x. A number equals itself. Symmetric Property if x = y, then y = x. Order of equality does not matter. Transitive Property if x = y and y = z, then x = z. Two numbers equal to the same number are equal to each other.

Properties of Congruence Reflexive Property AB ≅ AB A segment (or angle) is congruent to itself Symmetric Property If AB ≅ CD, then CD ≅ AB Order of equality does not matter. 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.

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

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

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 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. 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.

The Midpoint Theorem If M is the midpoint of AB, then AM = ½ AB and MB = ½ AB A M B

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

Statements Reasons 1. M is the midpoint of AB1. Given 2. AM  MB or AM = MB2. Definition of a midpoint 3. AM + MB = AB3. Segment Addition Postulate 4. AM + AM = AB4. Substitution Property Or 2 AM = AB 5. AM = ½ AB5. Division Property of Equality 6. MB = ½ AB6. Substitution Given: M is the midpoint of AB Prove: AM = ½ AB and MB = ½ AB A M B

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

1. BX is the bisector of  ABC;1. Given 2. m  ABX = m  BXC2. Definition of Angle Bisector or  ABX  m  BXC 3.m  ABX + m  BXC = m  ABC3. Angle Addition Postulate 4.m  ABX + m  ABX = m  ABC4. Substitution or 2 m  ABX = m  ABC 5. m  ABX = ½ m  ABC5. Division Property of Equality 6. m  BXC = ½ m  ABC6. Substitution Property Given: BX is the bisector of  ABC Prove: m  ABX = ½ m  ABC m  XBC = ½ m  ABC A B X C

Reasons Used in Proofs Given Information Definitions Postulates (including Algebra) Theorems

How to write a proof (The magical steps)

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, remember my magical purple pen….use yours.

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 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

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

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 See It!

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 See It!

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

Theorem Vertical angles are congruent

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.

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

A- Sometimes B – Always C - Never Right angles are ___________ vertical angles.

A- Sometimes B – Always C - Never Angles A, B, and C are __________ complementary.

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 = 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 = 75

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

White Board Practice Find the measure of a complement and a supplement of  T. m  T = 3x

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

2.5 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 angles are right. l m See It!

Theorem If two lines are perpendicular, then they form congruent, adjacent angles. l m 1 2 If l  m, then  1   2.

Theorem If two lines intersect to form congruent, adjacent angles, then the lines are perpendicular. l m 1 2 If  1   2, then l  m.

Theorem If the exterior sides of two adjacent angles lie on perpendicular lines, then the angles are complimentary. m l 1 2 If l  m, then  1 and  2 are compl. See It!

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

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

Remember Magical Proof Steps

Theorem If two angles are supplementary to congruent angles (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  

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

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