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Inductive Reasoning “Courage is resistance to fear, mastery of
fear, not absence of fear.” –Mark Twain
Inductive Reasoning
O Inductive Reasoning is the process of observing a
pattern and making a conjecture about the pattern.
O A conjecture is an unproven statement based on
observations.
O So, inductive reasoning is the process of observing
a pattern and making an unproven statement
about the pattern.
O Ex. 1: Describe the pattern and either draw the next
shape or write the next three numbers.
1) 5, 15, 45, 135, …..
1) 405, 1215, 3645
2) -2, 1, 4, 7, ……
10, 13, 16
3)
4)
O Ex. 2: Make and test conjectures for each problem below.
1) The sum of two odd numbers.
3+5=8 7+9=16 -5+7=2
The sum of two odd numbers is even.
1) The product of two even numbers.
4x6=24 8x8=64 -4x4=-16
The product of two even numbers is even.
1) The sum of three consecutive odd numbers.
3+5+7=3(5) 1+3+5=3(3) 15+17+19=3(17)
The sum of three consecutive odd numbers is three times the middle number.
Counter-example
O A counter-example is the case that shows a
conjecture to be false.
O Conjecture: The product of any two numbers
is always positive.
O Counter-example: -5 x 4 = -20
O Conjecture: We have school each weekday.
O Counter-example: October 10th
O Ex. 3: Find the counter example to each conjecture.
1) All prime numbers are odd.
1) 2
2) If the sum of two numbers is even, then both of
those numbers are also even.
1) 3+5=8
1) The sum of three consecutive numbers is always
odd.
1) 3+4+5=12
Summary
O You should now be able to:
O Identify patterns and make conjectures about
those patterns.
O Provide counter-examples to false
conjectures.
Conditional Statements
“The great use of life is to spend it for
something that will outlast it.” –William
James
Conditional statements
O Each conditional statement has a condition and a
consequence.
O Hypothesis: is the condition of the statement (also
the if portion of the if-then format).
O Conclusion: is the consequence of the statement
(also the then portion of the if-then format)
O Example: All mammals have hair.
O If-then form: If an animal is a mammal, then it has
hair.
Ex. 1: For each conditional statement, write it in if-then form.
1) Every student at Westfield has to take Physical Science
their freshman year.
1) If a student is a freshman at Westfield, then they have to
take Physical Science.
2) Two angles are complementary if their measures add up
to 90°.
1) If two angle measures sum to be 90°, then they are
complementary angles
3) Vertical angles have two pairs of opposite rays.
1) If a pair of angles are vertical angles, then they have two
pairs of opposite rays.
4) 2x+5=2, because x=-6
1) If x=-6, then 2x+5=2
Negation
O A negation is the opposite of the original
statement.
O The apple is red.
O Negation: the apple is NOT red.
Converse, Inverse, and Contrapositive.
O Conditional statement: If I forget to put my name on a paper, then I get a zero for that paper.
O Converse: switch the hypothesis and conclusion of the original conditional statement.
O If I get a zero for a paper, then I forget to put my name on the paper.
O Inverse: Negate BOTH the hypothesis and the conclusion of the original conditional statement.
O If I remember to put my name on a paper, then I will not get a zero for that paper.
O Contrapositive: Negate BOTH the hypothesis and the conclusion of the converse.
O If I get greater than a zero on a paper, then I remember to put my name on the paper.
Ex. 2: For each conditional statement, write its
converse, inverse, and contrapositive and decide each
statements truth value.
1) If you watch this video, then you take notes for
geometry.
1) Converse: If you take notes for geometry, then you
watch this video.
2) Inverse: If you don’t watch this video, then you don’t
take notes for geometry.
3) Contrapositive: If you don’t take notes for geometry,
then you don’t watch this video.
Equivalent and Biconditional
O Equivalent statements are statements that are both true or are both false. The conditional statement and contrapositive are always equivalent. The inverse and converse are always equivalent.
O Biconditional statements are statements that the original conditional statement and its converse are both true.
O For example: If two angle measures sum to be 90°, then they are complementary.
O The definition could be written as: Two angles are complementary if and only if (iff) their measures sum to 90°.
Ex. 3: For each conditional statement, write its
converse, inverse, and contrapositive and decide each
statements truth value. If both the conditional and its
converse are true, write a biconditional statement.
1) If four points are coplanar, then they lie in the same
plane.
1) Converse: If four points lie in the same plane, then
they are coplanar.
2) Inverse: If four points aren’t coplanar, then they don’t
lie in the same plane, .
3) Contrapositive: If four points don’t lie in the same
plane, then they aren’t coplanar.
4) Biconditional: Four points are coplanar iff they lie in
the same plane.
Summary
O At this point, you should be able to:
O Write a converse, inverse, and contrapositive
to a conditional statement.
O Know the requirements for a statement to be
biconditional.
O Know how to negate a statement.
Deductive Reasoning “Only the person who has faith in himself
is able to be faithful to others.”–Erich
Fromm
Deductive Reasoning
O Deductive reasoning uses facts, definitions,
properties, and laws of logic to form a logical
argument.
Laws of Logic
O Law of detachment: If the hypothesis of a
true conditional statement is true, then the
conclusion is also true.
O If the flipped method proves to improve
learning, then Mr. H will continue it for the
whole year.
O If Mr. H gets a haircut, then pigs can fly.
Law of Syllogism (dominoes)
For want of a nail the shoe was lost.
For want of a shoe the horse was lost.
For want of a horse the rider was lost.
For want of a rider the message was lost.
For want of a message the battle was lost.
For want of a battle the kingdom was lost.
O These statements could be combined to be:
For want of a nail, the kingdom was lost.
Law of Syllogism (dominoes)
O If q, then r.
O If r, then s.
O If q, then s.
Ex. 1: Use the Law of Detachment to make a valid
conclusion.
1) If two angles have the same measure, then they
are congruent. The measure of angle A is 90° and
the measure of angle B is 90°.
1) Angle A is congruent to angle B.
2) Pythagoras takes a nap at 4pm. It is 4pm on
Saturday.
1) Pythagoras is taking a nap.
Ex. 2: Use the Law of Syllogism to make a valid
conclusion.
1) If two angles are both right angles, then they have
the same measure. If two angles have the same
measure, then they are congruent.
1) If two angles are both right angles, then they are
congruent.
2) If Jesse get a job, then he can afford a car. If Jesse
can afford a car, then he buys a car.
1) If Jesse gets a job, then he buys a car.
Ex. 3: Determine whether each statement is the result
of inductive or deductive reasoning. Explain why.
1) For the last two weeks Mr. H has gone around
helping students during the class period. You
conclude that Mr. H will help students during the
class period on Monday.
1) Inductive because you are making a conjecture
based on previous observations.
2) The rule at work is that you have to work the full
week to get paid on Friday. You were paid on Friday.
Therefore, you went to all of your classes.
1) Deductive because you use rules and facts to make
a conclusion.
Summary
O You should be able to use the laws of logic
to make valid conclusions.
O You should be able to determine the
difference between deductive and inductive
reasoning.
Using Postulates and Diagrams
“A hero is no braver than an ordinary man
(or woman), but he (/she) is brave five
minutes longer.” –Ralph Waldo Emerson
Postulates 5-11
O 5: Through any two points there exists
exactly one line.
O 6: A line contains at least two points.
O 7: If two lines intersect, then their
intersection is exactly one point.
O 8: Through any three noncollinear points
there exists exactly one plane.
Postulates 5-11
O 9: A plane contains at least three
noncollinear points.
O 10: If two points lie in a plane, then the line
containing them lies in the plane.
O 11: If two planes intersect, then their
intersection is a line.
O Ex. 1: State the postulate illustrated by the
diagram.
1)
2) A A
C
A
B B
B
C
O Ex. 2: Use the diagram to write examples of
postulate 5 and 7.
Through points C and B there
is one line called line l.
Line DE and Line BF intersect
at point D.
O Ex. 3: Use the diagram to determine if each
statement is true or false.
1) Line AB lies in plane
R
2) Line FH lies in plane
R
3) Line AC and Line FG
will intersect.
4) Line GH is
perpendicular to
plane R.
5) Angle LGH is a right
angle.
6) Angle LGH and
angle LGF are
supplementary
angles.
L
Summary
O You should be able to identify the postulate
used in drawing a diagram.
Reasoning using Algebra.
“Certain signs precede certain events.”
–Cicero
Algebraic Properties of Equality.
Let a, b, and c be real
numbers.
1) Addition Property
2) Subtraction Property
3) Multiplication Property
4) Division Property
5) Substitution Property
6) Distributive Property
1) If 𝑎 = 𝑏, 𝑡ℎ𝑒𝑛 𝑎 + 𝑐 = 𝑏 + 𝑐.
2) 𝐼𝑓 𝑎 = 𝑏, 𝑡ℎ𝑒𝑛 𝑎 − 𝑐 = 𝑏 − 𝑐.
3) 𝐼𝑓 𝑎 = 𝑏, 𝑡ℎ𝑒𝑛 𝑎𝑐 = 𝑏𝑐.
4) 𝐼𝑓 𝑎 = 𝑏 𝑎𝑛𝑑 𝑐 ≠ 0, 𝑡ℎ𝑒𝑛 𝑎
𝑐=
𝑏
𝑐.
5) 𝐼𝑓 𝑎=𝑏, 𝑡ℎ𝑒𝑛 𝑎 𝑐𝑎𝑛 𝑏𝑒 𝑠𝑢𝑠𝑡𝑖𝑡𝑢𝑑𝑒𝑑 𝑖𝑛 𝑓𝑜𝑟 𝑎𝑛𝑦 𝑒𝑞𝑢𝑎𝑡𝑖𝑜𝑛 𝑜𝑟
𝑒𝑥𝑝𝑟e𝑠𝑠𝑖𝑜𝑛.
6) 𝑎 𝑏 + 𝑐 = 𝑎𝑏 + 𝑎𝑐
Example 1: Solve 2𝑥 + 30 = 75 − 3𝑥. Write a
reason for each step.
Equation
2𝑥 + 30 = 75 − 3𝑥
5𝑥 + 30 = 75
5𝑥 = 45
𝑥 = 9
Reason
Given
Addition Property
Subtraction Property
Division Property
Example 2: Solve −2(𝑥 + 30) = 2(70 − 3𝑥). Write a
reason for each step.
Equation
−2(𝑥 + 30) = 2(70 − 3𝑥)
−2𝑥 − 60 = 140 − 6𝑥
4𝑥 − 60 = 140
4𝑥 = 200
𝑥 = 50
Reason
Given
Distributive Property
Addition Property
Addition Property
Division Property
Reflexive Properties of Equality.
1) Real Numbers
2) Segment Length
3) Angle Measure
1) 𝐹𝑜𝑟 𝑎𝑛𝑦 𝑟𝑒𝑎𝑙 𝑛𝑢𝑚𝑏𝑒𝑟 𝑎, 𝑎 = 𝑎.
2) 𝐹𝑜𝑟 𝑎𝑛𝑦 𝑠𝑒𝑔𝑚𝑒𝑛𝑡 𝐴𝐵, 𝐴𝐵 = 𝐴𝐵
3) 𝐹𝑜𝑟 𝑎𝑛𝑦 𝑎𝑛𝑔𝑙𝑒 𝐴, 𝑚∠𝐴 = 𝑚∠𝐴.
Symmetric Properties of Equality.
1) Real Numbers
2) Segment Length
3) Angle Measure
1) 𝐹𝑜𝑟 𝑎𝑛𝑦 𝑟𝑒𝑎𝑙 𝑛𝑢𝑚𝑏𝑒𝑟 𝑎 𝑎𝑛𝑑 𝑏, 𝑎 = 𝑏, 𝑡ℎ𝑒𝑛 𝑏 = 𝑎.
2) 𝐹𝑜𝑟 𝑎𝑛𝑦 𝑠𝑒𝑔𝑚𝑒𝑛𝑡 𝐴𝐵 𝑎𝑛𝑑 𝐶𝐷, 𝑖𝑓 𝐴𝐵 = 𝐶𝐷, 𝑡ℎ𝑒𝑛 𝐶𝐷 = 𝐴𝐵.
3) 𝐹𝑜𝑟 𝑎𝑛𝑦 𝑎𝑛𝑔𝑙𝑒 𝐴 𝑎𝑛𝑑 𝐵, 𝑖𝑓 𝑚∠𝐴 = 𝑚∠𝐵, 𝑡ℎ𝑒𝑛 𝑚∠𝐵 = 𝑚∠𝐴.
Transitive Properties of Equality.
1) Real Numbers
2) Segment Length
3) Angle Measure
1) 𝐼𝑓 𝑎 = 𝑏 𝑎𝑛𝑑 𝑏 = 𝑐, 𝑡ℎ𝑒𝑛 𝑎 = 𝑐.
2) 𝐼𝑓𝐴𝐵 = 𝐶𝐷 𝑎𝑛𝑑 𝐶𝐷 = 𝐸𝐹, 𝑡ℎ𝑒𝑛 𝐴𝐵 = 𝐸𝐹.
3) 𝐼𝑓 𝑚∠𝐴 = 𝑚∠𝐵 𝑎𝑛𝑑 𝑚∠𝐵 = 𝑚∠𝐶, 𝑡ℎ𝑒𝑛 𝑚∠𝐴 = 𝑚∠𝐶.
Example 3: Determine if 𝑚∠ABC= 𝑚∠𝐹𝐵𝐷. Show your reasoning.
Equation
𝑚∠1= 𝑚∠3
𝑚∠ABC= 𝑚∠1+ 𝑚∠2
𝑚∠FBD= 𝑚∠2 +𝑚∠3
𝑚∠FBD= 𝑚∠2+ 𝑚∠1
𝑚∠ABC= 𝑚∠FBD
Reason
Given
Angle Addition Postulate
Angle Addition Postulate
Substitution Property
Transitive Property
A
B
C
D
F
1 2
3
Summary
O You should be able to use properties to
justify your reasoning.
O You should be able to solve a problem and
provide reasons for each step.
Prove Statements about segments and
angles “Anxiety is fear of one’s self.” –Wilhelm
Stekel
Congruence of Segments and Angles. (Theorem 2.1 and 2.2)
1) Reflexive Property of Congruence
2) Symmetric Property of Congruence
3) Transitive Property of Congruence
For any segment AB and any angle A.
1) 𝐴𝐵 ≅ 𝐴𝐵 or ∠𝐴 ≅ ∠𝐴
2) If 𝐴𝐵 ≅ 𝐶𝐷, 𝑡ℎ𝑒𝑛 𝐶𝐷 ≅ 𝐴𝐵 or If ∠𝐴 ≅ ∠𝐵, 𝑡ℎ𝑒𝑛 ∠𝐵 ≅ ∠𝐴
3) 𝐼𝑓 𝐴𝐵 ≅ 𝐶𝐷 𝑎𝑛𝑑 𝐶𝐷 ≅ 𝐸𝐹, 𝑡ℎ𝑒𝑛 𝐴𝐵 ≅ 𝐸𝐹 or
𝐼𝑓 ∠𝐴 ≅ ∠𝐵 𝑎𝑛𝑑 ∠𝐵 ≅ ∠𝐶, 𝑡ℎ𝑒𝑛 ∠𝐴 ≅ ∠𝐶.
Example 1: Use a two column proof to show that AC ≅ 𝐵𝐷.
A B C D
Statements
1) 𝐴𝐵 = 𝐶𝐷
2) 𝐴𝐶 = 𝐴𝐵 + 𝐵𝐶
3) 𝐵𝐷 = 𝐶𝐷 + 𝐵𝐶
4) 𝐵𝐷 = 𝐴𝐵 + 𝐵𝐶
5) 𝐴𝐶 = 𝐵𝐷
6) AC ≅ 𝐵𝐷
Reasons
1) Given
2) Segment Addition Postulate
3) Segment Addition Postulate
4) Substitution Property
5) Transitive Property
6) Definition of Congruence
Ex. 2: Name the property illustrated by the statement.
1) 𝐼𝑓 ∠𝐹 ≅ ∠𝐺 𝑎𝑛𝑑 ∠𝐺 ≅ ∠𝐻, 𝑡ℎ𝑒𝑛 ∠𝐹 ≅ ∠𝐻.
2) If 𝐸𝐹 ≅ 𝐺𝐻, 𝑡ℎ𝑒𝑛 𝐺𝐻 ≅ 𝐸𝐹.
3) 𝐴𝐵 ≅ 𝐴𝐵
Example 1:Prove that AB=2𝐴𝑀. 𝑌𝑜𝑢 𝑘𝑛𝑜𝑤 𝑡ℎ𝑎𝑡 𝑀 𝑖𝑠 𝑡ℎ𝑒
𝑚𝑖𝑑𝑝𝑜𝑖𝑛𝑡 𝑜𝑓 𝐴𝐵.
A M B
Statements
1) 𝑀 𝑖𝑠 𝑎 𝑚𝑖𝑑𝑝𝑜𝑖𝑛𝑡 𝑜𝑓 𝐴𝐵. 2) 𝐴𝑀 ≅ 𝑀𝐵
3) 𝐴𝑀 = 𝑀𝐵 4) 𝐴𝐵 = 𝐴𝑀 + 𝑀𝐵
5) 𝐴𝐵 = 𝐴𝑀 + 𝐴𝑀
6) 𝐴𝐵 = 2𝐴𝑀
Reasons
1) Given
2) Definition of Midpoint
3) Definition of congruence
4) Segment Addition Postulate
5) Substitution Property
6) Simplify
Summary
O You should be able to prove statements
about segments and angles.
O You should be able to write a two column
proof.
Prove Angle Pair Relationships.
“Remember that happiness is a way of
travel—not a destination.” –Roy M.
Goodman
Theorem 2.3-5
O Theorem 2.3: Right Angles Congruence Theorem.
O All right angles are congruent.
O Theorem 2.4: Congruent Supplements Theorem.
O If two angles are supplementary to the same angle
(or to congruent angles), then they are congruent.
O Theorem 2.5: Congruent Complements Theorem.
O If two angles are complementary to the same angle
(or to congruent angles), then they are congruent.
1) Ex. 1: Prove that ∠1 ≅ ∠3, given that ∠1 𝑎𝑛𝑑 ∠2 are
supplementary and ∠3 𝑎𝑛𝑑 ∠2 are supplementary.
2
3 1
Statements
1) ∠1 𝑎𝑛𝑑 ∠2 are supp.
∠3 𝑎𝑛𝑑 ∠2 are supp.
2) 𝑚∠1 + 𝑚∠2 = 180
3) 𝑚∠3 + 𝑚∠2 = 180 4) 𝑚∠1 + 𝑚∠2 = 𝑚∠3 + 𝑚∠2
5) 𝑚∠1 = 𝑚∠3
6) ∠1 ≅ ∠3
Reasons
1) Given
2) Definition of Supplementary
3) Definition of Supplementary
4) Transitive Property
5) Subtraction Property
6) Definition of Congruence
Postulate 12 and Theorem 2.6
O Postulate 12: Linear Pair Postulate.
O If two angles form a linear pair, then they are
supplementary.
O Theorem 2.6: Vertical Angles Congruence Theorem.
O Vertical Angles are Congruent.
O Ex. 2: Given that angle 3 and angle 4 are a linear
pair and measure of angle 4 is 112°, find the
measure of angle 3.
Statements
1) ∠3 𝑎𝑛𝑑 ∠4 form a linear
pair and 𝑚∠4 = 112°
2) ∠3 𝑎𝑛𝑑 ∠4 are supp.
3) 𝑚∠3 + 𝑚∠4 = 180
4) 𝑚∠3 + 112 = 180 5) 𝑚∠3 = 68°
Reasons
1) Given
2) Linear Pair Postulate
3) Definition of Supplementary
4) Substitution Property
5) Subtraction Property
Ex. 3: Find the value of x if 𝑚∠1 = (3𝑥 − 4)° and 𝑚∠4 = (6𝑥 − 184)°.
1
2 3
4 5
Statements
1) 𝑚∠1 = 3𝑥 − 4 ° 2) 𝑚∠4 = (6𝑥 − 184)°
3) ∠1≅∠4
4) 𝑚∠1 = 𝑚∠4
5) 3𝑥 − 4 ° = (6𝑥 − 184)°
6) −4 = 3𝑥 − 184 7) 180=3x 8) 60=x
Reasons
1) Given
2) Given
3) Vertical Angle Congruence
Theorem
4) Definition of Congruence
5) Transitive Property
6) Subtraction Property
7) Addition Property
8) Division Property
Summary
O You should be able to identify
complementary and supplementary angles.
O You should be able to identify linear pairs
and vertical angles.
O You should be able to use the above
definitions, postulates, and theorems to
write a proof.