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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
Transitive Property Reflexive Property
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
6) 6) 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)
5) 5) Segment Addition Postulate
6) 6) Segment Addition Postulate
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)
3) 3) Segment Addition Postulate
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
Transitive Property Reflexive Property
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
4) 4) Angle Addition Postulate
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
3) 3) Angle Addition Postulate
4) 4) Angle Addition Postulate
5) 5) Reflexive Property
6) 6) Definition of Congruent Angles
7) 7) Addition Property of Equality
8) 8) Substitution
9) 9) Definition of Congruent Angles
4) Given: ∠𝐴𝐸𝐶 ≅ ∠𝐵𝐸𝐷
Prove: ∠𝐴𝐸𝐵 ≅ ∠𝐶𝐸𝐷
Statements Reasons
1) 1) Given
2) 2)
3) 3) Angle Addition Postulate
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
3) 3) Angle Addition Postulate
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)