Day 1 Inductive Reasoning and Conjectures · Formal Geometry Chapter 2 – Logic and Proofs Day 1...

Formal Geometry Chapter 2 – Logic and Proofs

Day 1 – Inductive Reasoning and Conjectures

Objectives: SWBAT form a conjecture, and check it

SWBAT use counterexamples to disprove a conjecture

Logic the use of valid reasoning to justify a claim

Inductive Finding patterns, experiments in information to

justify a claim.

Deductive Using laws, facts, or previously proven truths to

justify a claim.

Conjecture Something you think is true; an observation – can be true

or false Example: Make a conjecture for the following.

1. 3, 5, 9, 15, 23….. add the next even number in the sequence______

2. 3+5=8, 5+1=6, 11+3=14, 7+13=20 _add two odds #s to get an even #_

3. __Add a side_____

For a conjecture to be truth, it has to be truth in ALL Cases! Although this is impossible, we do our best and use deductive reasoning to help.

Counterexample

One instance when a conjecture is false

Find a counter example to the following conjectures.

1. When you add a positive number and a negative number, the answer is

negative.

2. All bears are brown or black.

Polar Bears or Gummi Bears

3. All jokes are hilariously funny.

What happens when you hurt your toe? You call a tow truck =D

Write a conjecture that describes the pattern in each sequence. Then make a conjecture for the next term.

8 12 4

10,000 100 9,900

45

12 30

Add min

:

2 25 11 25Add $ . $ .

Sha de t o

the Left

Add another

circle

Day 2 – Conditional Statements

Objectives: SWBAT Recognize and analyze a conditional statement

Conditional Statement

An “if_______ then ________” statement

If-then form must be put into this form, even if it isn’t to start

Hypothesis the first clause (without the “if”)

Conclusion the second clause (without the “then”)

Rewrite the conditional statements in if-then form.

1. Two points are collinear if they lie on the same line.

If two points are collinear, then they lie on the same line.

2. An obtuse angle is an angle that measures more than 90 and less than 180

If an angle is obtuse, then it measures more than 90 and less than 180

3. Cheese contains Calcium.

If it is cheese, then it contains calcium

Identify the hypothesis and conclusion of each conditional statement.

1. If today is Friday, then tomorrow is Saturday.

2. If two angles are supplementary, then they add up to 180 degrees.

Inverse The negation of the Conditional statement. Think adding

words like not, or isn’t.

Converse Switching the order of the Conditional Statement. The If

and Then stay put but the Hypo and Conclusion switch.

Contrapositive The negation of the Converse

Bi-conditional “If and only if” must be true backwards and

forwards

Hypo

Hypo

Conclusion

Conclusion

Write the following statements in if-then form. Then write the converse, contrapositive, inverse, and biconditional statements.

If there is fresh snow on the mountains, then it is a great day for skiing.

Inverse:

If there is not fresh snow on the mountains, then it is not a great day for skiing.

Converse:

If it is a great day for skiing, then there is fresh snow on the mountains.

Contrapositive:

If it is not a great day for skiing, then there is not fresh snow on the mountains.

Bi-conditional: There is fresh now on the mountains if and only if it is a great day for skiing. Write the converse, contrapositive, inverse, and biconditional statements.

If today is Saturday, then we do not have school today. Converse: If we do not have school today, then today is Saturday.

Inverse: If today is not Saturday, then we do have school today.

Contrapositive: If we do have school today, then today is not Saturday.

Bi-conditional: Today is Saturday if and only if we do not have school today.

Fun Facts Conditional Statements and Contrapositives If the Conditional statement is true, then the Contrapositive is also true.

Converses and Inverses

If the converse is true, then the inverse is also true.

Day 3 – Deductive Reasoning and Laws of Logic

Objectives: SWBAT form a conjecture, and check it

SWBAT use counterexamples to disprove a conjecture

Logic

Inductive Finding patterns, experiments in information to

justify a claim.

Deductive Using laws, facts, or previously proven truths to

justify a claim.

Laws of Logic

The Law of Detachment

If you have a true compound statement p q. If you then prove p to be true, then you can conclude that q is also true.

The Law of Contrapositive If you have a true compound statement pq. If you then prove ~q to be true, then you can conclude that ~p is also true.

The Law of Syllogism

Train Property

...

...

If

p q is true

p is true

then

q is true

...

~

...

~

If

p q is true

q is true

then

p is true

...

...

...

If

p q

and

q r

then

p r

Determine whether each conclusion is based on inductive or deductive reasoning.

1. Students at Reno High School must have a B average in order to participate in sports. Hank has a B average, so he concludes that he can play sports at Reno

High.

Deductive Reasoning

2. Holly notices that every Saturday, her neighbor mows his lawn. Today is Saturday. Holly concludes that her neighbor will mow his lawn.

Inductive Reasoning

Determine whether the stated conclusion is valid based on the given information. If not, write invalid. Explain your reasoning.

3. Given: If a number is divisible by 4, when the number is also divisible by 2.

Claim: 12 is divisible by four.

Conclusion: 12 is divisible by two.

True Statement – Law of Detachment

4. If Edward stays up late, he will be tired the next day Claim: Edward is tired.

Conclusion: Edward stayed up late.

False Statement – he could be tired for lots of reasons.

5. If you study, then you are prepared for the Celebration of Knowledge. If you are

prepared for the Celebration of Knowledge, you won’t panic. If you won’t panic, then you will get a good grade. Claim: Stanley studied for the Celebration

Conclusion: Stanley will get a good grade.

True Statement – Law of Syllogism

Two Points Make a Line Postulate Through any two points is exactly one line.

Three Non-Collinear Points Postulate Any 3 noncollinear points will make exactly one plane.

Lines on Planes Postulate If two points lie on a plane, then the entire line containing those points lies in that

plane.

Intersecting Lines Postulate

If two lines intersect, then their intersection is exactly one point.

Intersecting Planes Postulate If two planes intersect, then their intersection is a line.

Inter secting

Planes Postulate

3

Non Collinear

Point s Postulate

Sometimes

Could be a point or line

2

Always

Point s Make A line Postulate

2

Always

Point s Make A line Postulate

Day 4 – Algebraic Proofs

Objectives: SWBAT form a proof using Algebra

Addition Property of Equality If a = b, then a + c = b + c

Subtraction Property of Equality

if a = b, then a – c = b – c

Multiplication Property of Equality

if a = b, then a x c = b x c

Division Property of Equality

if a = b, then 𝒂

𝒄=

𝒃

𝒄

Distributive Property

~ 𝒂(𝒃 + 𝒄) = 𝒂𝒃 + 𝒂𝒄

Math Fact / Simplifying Like Terms Combining like terms on the same side of an equals sign

Proof Basics: Given: Information given that does not need to be proved true

Prove: Information that is needed to be proven true – the final goal of a proof

Examples

1. Given: 𝟏𝟎𝒚 + 𝟓 = 𝟐𝟓

Prove: 𝒚 = 𝟐

10 5 25y Given

10 20y Subtraction Property of Equality

2y Division Property of Equality

Given: 𝟔𝒙 + 𝟑 = 𝟗(𝒙–𝟏) Prove: 𝒙 = 𝟒

6 3 9 1x x Given

6 3 9 9 x x Distributive Property

3 3 9 x Subtraction Property of Equality

12 3 x Addition Property of Equality

4x Division Property of Equality

Given: 𝟔𝒙 + 𝟕 = 𝟖𝒙 − 𝟓 Prove: 𝒙 = 𝟔

6 7 8 5 x x Given

7 2 5 x Subtraction Property of Equality

12 2 x Addition Property of Equality

6x Division Property of Equality

Given: 𝟏

𝟓𝒎 + 𝟑 = 𝟐𝒎 − 𝟐𝟒 Prove: 𝒎 = 𝟏𝟓

13 2 24

5 m m

Given

15 10 120 m m Multiplication Property of Equality

15 9 120 m Subtraction Property of Equality

145 9 m Addition Property of Equality

15 m Symmetric Property

15m Division Property of Equality

5. Given: 𝟐𝒎 = 𝒏 + 𝟓 𝒎 = 𝒏 − 𝟏

Prove: 𝒏 = 𝟕

𝟐𝒎 = 𝒏 + 𝟓 Given

𝒎 = 𝒏 − 𝟏 Given

𝟐(𝒏 − 𝟏) = 𝒏 + 𝟓 Substitution

𝟐𝒏 − 𝟐 = 𝒏 + 𝟓 Distributive Property

𝟐𝒏 = 𝒏 + 𝟕 Addition Property of Equality

𝒏 = 𝟕 Subtraction Property of Equality

Day 5 - Introduction to Proofs

Objectives: SWBAT form a proofs

What’s a proof? What makes up a proof? What can be used for the reasons?

Can Be Assumed

Cannot Be Assumed

Coplanar Points Perpendicular Lines

Collinear Points Complementary Angles

Betweenness of Points Congruent Angles

Intersection Points Congruent Segments

Interior / Exterior of Angles

Straight Lines (Linear pairs)

Vertical Angles

Examples:

1. Given: B is the midpoint of AC .

Prove: AB BC .

Statements Reasons

A B C

2. Given: BD bisects ABC

Prove: ABD DBC

Statements Reasons

3. Given: m 1 = 55 .

Prove: m 3 55

Statements Reasons

Reflexive Property

Symmetric Property

Substitution Property

Transitive Property

A

B

C

D

1 23

4

Day 7 – Segment Proofs – Part 1

Objectives: SWBAT form a proofs involve Segments and Segment Addition

Key Postulates / Theorems to Remember……

Definition of Congruent Segments

Segment Bisector Definition of Midpoint

The first Statement and Reason for a proof is _________________________.

The last statement of a proof is always ________________________________.

When do Segments get hats? __________________________________________. Segment Proofs: fill in the blanks of the following segment proofs.

1. Given: Y is the midpoint of XZ .

Prove: XY YZ

Statements Reasons

1) 1) Given

2) XY YZ 2)

2. Given: M is the midpoint of LP .

Prove: 𝐿𝑀 = 𝑀𝑃

Statements Reasons

1) 1) Given

2) DE EF 2)

3) 3)

3. Given: 𝐴𝐵̅̅ ̅̅ 𝑏𝑖𝑠𝑒𝑐𝑡𝑠 𝐷𝐹̅̅ ̅̅

Prove: 𝐷𝐸 = 𝐸𝐹

Statements Reasons

1) 1) Given

2) 2)

3) 𝐷𝐸 = 𝐸𝐹 3)

Transitive Property Reflexive Property

4. Given: ,HJ KL KL MN

Prove: HJ MN

Statements Reasons

1) 1) Given

2) 2) Given

3) HJ MN 3)

4. Given: , , 11XY WZ WZ ZY ZY

Prove: 11XY

Statements Reasons

1) 1) Given

2) 2) Given

3) 3) Given

4) 4)

5) 5)

6) 11XY 6)

6.

Statements Reasons

1) 1) Given

2) 2) Given

3) 3) Given

4) AB BC 4)

5) 5) Definition of Midpoint

6) BC EF 6)

Given :

B is the midpoint of AC

E is the midpoint of DF

Prove :

AB DE

BC EF

Day 8 – Segment Proofs – Part 2

Objectives: SWBAT form a proofs involve Segments and Segment Addition

Segment Addition Postulate Definition of Congruent Segments


Substitution Property Definition of Congruent Segments

1. Given: Diagram

Prove: EF FG EG

Statements Reasons

1) 1) Given

2) EF FG EG 2)

2. Given: 𝑇𝑈 = 14, 𝑈𝑉 = 21

Prove: 𝑇𝑉 = 35

Statements Reasons

1) 1) Given

2) 2) Given

3) 3)

4) 4)

5) 5)

3. Given: 𝐴𝐵 = 𝑥, 𝐵𝐶 = 𝑦

Prove: 𝐴𝐶 = 𝑥 + 𝑦

Statements Reasons

1) 1) Given

2) 2) Given

3) 3)

4) 4)

Segment Addition Proof Pattern

4. Given:, MP EG , NP EF

Prove: MN FG

Statements Reasons

1) 1) Given

2) 2) Given

3) MP EG 3)

4) NP EF 4)

5) 5) Segment Addition Postulate


7) MN NP EF FG 7)

8) MN FG 8)

9) MN FG 9)

5. Given: 𝑋𝑌̅̅ ̅̅ ≅ 𝐵𝐶̅̅ ̅̅ , 𝑌𝑍̅̅̅̅ ≅ 𝐴𝐵̅̅ ̅̅

Prove: 𝑋𝑍̅̅ ̅̅ ≅ 𝐴𝐶̅̅ ̅̅

Statements Reasons

1) 1) Given

2) 2) Given

3) XY AB 3)

4) YZ AB 4)



7) XY YZ AB BC 7)

8) XZ AC 8)

9) XZ AC 9)

6. Given: HJ KL

Prove: HK JL

Statements Reasons

1) 1) Given

2) HJ KL 2)


4) HJ JK HK 4)

5) JK JK 5)

6) 6) Definition of Congruent Segments

7) JK HJ JK KL 7)

8) 8) Substitution

9) 9)

10) HK JL 10)

7. Given: , AB BC BC CD

Prove: AC BD

Statements Reasons

1) 1) Given

2) 2) Given

3) AB BC 3)

4) BC CD 4)

5) AC AB BC 5)

6) BC BC CD 6)

7) 7) Reflexive Property

8) 8) Definition of Congruent Segments

9) 9) Addition Prop of =

10) 10) Substitution

11) AC BD 11)

Day 9 – Angle Proofs – Part 1

Objectives: SWBAT form a proofs involve Angle Addition

Angle Addition Postulate Definition of Congruent Segments


Substitution Property Definition of Congruent Angles

1) Given: 𝑚∠𝑅𝑆𝑇 = 50° 𝑚∠𝑇𝑆𝑉 = 40°

Prove: 𝑚 < 𝑅𝑆𝑉 = 90°

Statements Reasons

1) 1) Given

2) 2) Given

3) 3) Angle Addition Postulate

4) 𝑚 < 𝑅𝑆𝑉 = 40 + 50 4)

5) 𝑚 < 𝑅𝑆𝑉 = 90° 5)

2) Given: 𝑚∠1 = 20° 𝑚∠2 = 40° 𝑚∠3 = 30°

Prove: 𝑚 < 𝑋𝑌𝑍 = 90°

Statements Reasons

1) 1) Given

2) 𝑚∠2 = 40° 2)

3) 3) Given


5) 𝑚 < 𝑋𝑌𝑍 = 20 + 40 + 30 5)

6) 𝑚 < 𝑋𝑌𝑍 = 90° 6)

3) Given: ∠𝐻𝐺𝐽 ≅ ∠𝐾𝐺𝐿

Prove: ∠𝐻𝐺𝐾 ≅ ∠𝐽𝐺𝐿

Statements Reasons

1) 1) Given

2) 2) Definition of Congruent Angles



5) 5) Reflexive Property


7) 7) Addition Property of Equality

8) 8) Substitution


4) Given: ∠𝐴𝐸𝐶 ≅ ∠𝐵𝐸𝐷

Prove: ∠𝐴𝐸𝐵 ≅ ∠𝐶𝐸𝐷

Statements Reasons

1) 1) Given

2) 2)


4) 4)

5) 5) Substitution

6) 6) Subtraction Property of Equality

7) 7)

5) Given: ∠1 ≅ ∠4

∠2 ≅ ∠3

Prove: ∠𝐴𝐵𝐶 ≅ ∠𝐷𝐸𝐹

Statements Reasons

1) ∠1 ≅ ∠4 1)

2) 2)

3) 𝑚∠1 = 𝑚∠4 3)

4) 4)

5) 𝑚∠1 + 𝑚∠2 = 𝑚∠𝐴𝐵𝐶 5)

6) 6)

7) 𝑚∠1 + 𝑚∠2 = 𝑚∠𝐷𝐸𝐹 7)

8) 8)

9) ∠𝐴𝐵𝐶 ≅ ∠𝐷𝐸𝐹 9)

Day 10 – Angle Proofs – Part 2

Objectives: SWBAT form a proofs involve Angles and Angle Addition

Angle Bisector Definition of Complementary Angles

Linear Pair Definition of Supplementary Angles

Vertical Angles Right Angle Theorem

Angle Proofs: fill in the blanks of the following angle proofs.

1) Given: 𝑚∠𝐶𝐷𝐸 = 110°

𝑚∠𝐹𝐺𝐻 = 110°

Prove: ∠𝐶𝐷𝐸 ≅ ∠𝐹𝐺𝐻

Statements Reasons

1) 1) Given

2) 2) Given

3) 3) Substitution / Transitive Property

4) ∠𝐶𝐷𝐸 ≅ ∠𝐹𝐺𝐻 4)

2) Given: 𝑚∠𝑅𝑆𝑇 = 50° 𝑚∠𝑇𝑆𝑉 = 40°

Prove: < 𝑅𝑆𝑇 𝑎𝑛𝑑 < 𝑇𝑆𝑉 𝑎𝑟𝑒 𝐶𝑜𝑚𝑝𝑙𝑒𝑚𝑒𝑛𝑡𝑎𝑟𝑦

Statements Reasons

1) 𝑚∠𝑅𝑆𝑇 = 50° 1)

2) 2) Given


4) 𝑚 < 𝑅𝑆𝑉 = 40 + 50 4)

5) 𝑚 < 𝑅𝑆𝑉 = 90° 5)

6) < 𝑅𝑆𝑇 𝑎𝑛𝑑 < 𝑇𝑆𝑉 𝑎𝑟𝑒 𝐶𝑜𝑚𝑝𝑙𝑒𝑚𝑒𝑛𝑡𝑎𝑟𝑦 6)

3) Given: 𝑚∠𝐶𝐷𝐸 = 110° 𝑚∠𝐹𝐺𝐻 = 70°

Prove: < 𝐶𝐷𝐸 𝑎𝑛𝑑 < 𝐹𝐺𝐻 𝑎𝑟𝑒 𝑆𝑢𝑝𝑝𝑙𝑒𝑚𝑒𝑛𝑡𝑎𝑟𝑦

Statements Reasons

1) 𝑚∠𝐶𝐷𝐸 = 110° 1)

2) 𝑚∠𝐹𝐺𝐻 = 70° 2)

3) 110 + 70 = 180 3) Math Fact

4) 𝑚∠𝐶𝐷𝐸 + 𝑚∠𝐹𝐺𝐻 = 180 4)

5) < 𝐶𝐷𝐸 𝑎𝑛𝑑 < 𝐹𝐺𝐻 𝑎𝑟𝑒 𝑆𝑢𝑝𝑝𝑙𝑒𝑚𝑒𝑛𝑡𝑎𝑟𝑦 5)

4) Given: 𝑊𝑌⃗⃗⃗⃗⃗⃗ ⃗ 𝑏𝑖𝑠𝑒𝑐𝑡𝑠 < 𝑋𝑊𝑍

Prove: 𝑚∠1 = 𝑚∠3

Statements Reasons

1) 1) Given

2) 𝒎∠𝟏 = 𝒎∠𝟐 2)

3) 3) Vertical Angles Theorem

4) 𝒎∠𝟏 = 𝒎∠𝟑 4)

5) 5) Substitution / Transitive Property

5. Given: ∠1 ≅ ∠2, ∠1 ≅ ∠4

Prove: 𝐹𝐻̅̅ ̅̅ 𝑏𝑖𝑠𝑒𝑐𝑡𝑠 < 𝐸𝐹𝐺

Statements Reasons

1) 1)

2 2)

3) 3)

4) 4)

6. Given: A D

supp.A and B are

supp.C and D are

Prove: B C

Statements Reasons

1) 1)

2) 2)

3) 3)

4) 4)

5) 5)

7. Given: ∠𝑉 ≅ ∠𝑌𝑅𝑋, ∠𝑌 ≅ ∠𝑇𝑅𝑉

Prove: ∠𝑉 ≅ ∠𝑌

Statements Reasons

1) 1)

2) 2)

3) 3)

4) 4)

Day 1 Inductive Reasoning and Conjectures · Formal Geometry Chapter 2 – Logic and Proofs Day 1...

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