Transcript 2.5: Conjectures That Lead to Theorems

2.5: Conjectures That Lead to
Theorems
Expectations:
G1.1.1: Solve multistep problems and construct proofs involving vertical
angles, linear pairs of angles, supplementary angles,
complementary angles and right angles.
G3.1.3: Find the image of a figure under the composition of two or more
isometries and determine whether the resulting figure
is a reflection, rotation, translation, or glide reflection image of the
original figure.
L3.1.1: Distinguish between inductive and deductive reasoning,
identifying and providing examples of each.
L3.1.2: Differentiate between statistical arguments (statements verified
empirically using examples or data) and logical arguments based
on the rules of logic.
L3.3.1: Know the basic structure for the proof of an “if…, then” statement
and that proving the contrapositive is equivalent.
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2.5: Conjectures that Lead to Theorems
Inductive Reasoning
Inductive reasoning is based on
observations or patterns.
Inductive reasoning is NOT valid for
proofs.
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Inductive Reasoning
Betty observed 5 white cars traveling
very slowly down the road so she
concludes that all white cars are slow.
Betty used inductive reasoning to reach
her conclusion.
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2.5: Conjectures that Lead to Theorems
Deductive Reasoning
Deductive reasoning is based on known
facts, or statements such as
postulates definitions or theorems.
Deductive reasoning is valid for proofs.
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2.5: Conjectures that Lead to Theorems
Deductive Reasoning
Triangle ABC is a right triangle so Billy
concludes triangle ABC has a right angle.
Billy has used deductive reasoning because
the definition of a right triangle tells him
that it has a right angle.
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2.5: Conjectures that Lead to Theorems
Which of the following statements demonstrates
deductive reasoning?
A. All crows are black, and all crows are birds; therefore, all
birds are black.
B. All dolphins have fins, and all fish have fins; therefore,
all dolphins are fish.
C. Edward is a human being, and all human beings are
mortal; therefore, Edward is mortal.
D. Megan gets good grades, and studying results in good
grades; therefore, Megan is studying.
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2.5: Conjectures that Lead to Theorems
Statistical Arguments vs Logical
Arguments
A statistical argument is made using _____ or
___________ to justify your statements.
A logical argument is made by combining true
statements (postulates, definitions and
theorems) together to reach a conclusion.
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2.5: Conjectures that Lead to Theorems
Is this a statistical argument or a
logical argument?
Barney read in his owners manual that he can
increase his gas mileage by 15% if he slows
down by an average of 10 miles per hour on the
expressway. He then suggests to his brother
that he too should slow down to save gas.
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2.5: Conjectures that Lead to Theorems
Is Pebbles using a statistical
argument or a logical argument?
Given A, B and C are collinear points and that A
and B are both on plane Q, Betty is trying to
determine if C must also be on Q. Pebbles says
C must be on Q because of the Unique Plane
Postulate.
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2.5: Conjectures that Lead to Theorems
Who is using a statistical
argument?
Harry picked 3 cards out of a deck of cards and selected 3 hearts.
Ron observed this and said the deck must not be a regular deck of
cards because there is no way you can draw 3 straight hearts from
a deck of cards.
Hermione said it could be a regular deck because there is about a 1%
chance of drawing 3 in a row of any suit.
A. only Ron
B. only Hermione
C. only Harry
D. Ron and Hermione
E. no one is using a statistical argument
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2.5: Conjectures that Lead to Theorems
Congruent Supplements Theorem:
If 2 angles are supplements of congruent angles
(or the same angle), then they are __________.
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2.5: Conjectures that Lead to Theorems
Congruent Supplements Theorem:
1
2
4
3
If ∠1 ≅ ∠ 3, ∠ 1 is supplementary ∠ 2 and ∠ 3
is supplementary ∠ 4, then ____________.
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2.5: Conjectures that Lead to Theorems
Vertical Angles
Defn: Two angles are vertical angles iff they are
the nonadjacent angles formed by two
intersecting lines.
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3
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2.5: Conjectures that Lead to Theorems
Vertical Angles
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1
3
2
Make a conjecture about vertical angles.
Try to justify your conjecture with mathematical
statements.
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2.5: Conjectures that Lead to Theorems
Vertical Angle Theorem
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2.5: Conjectures that Lead to Theorems
Prove the Vertical Angle Theorem
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2.5: Conjectures that Lead to Theorems
Determine the measure of ∠1
x2 + 2x +4
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1
3
2x2 + 5x -50
2.5: Conjectures that Lead to Theorems
Complete Activity 2 on page 119.
You may work in pairs.
You have 10 minutes.
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2.5: Conjectures that Lead to Theorems
Two Reflections Theorem for
Translations.
If a transformation is the composite of two
reflections over parallel lines, then it is
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You want to translate ΔABC 10 units by reflecting
it twice. Describe as accurately as you can how
to position the lines.
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Two Reflections Theorem For
Rotations
If a transformation is the composite of two
reflections over intersecting lines, then it is
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Two Reflections Theorem for
Rotations
l
F”
70
°
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m
2.5: Conjectures that Lead to Theorems
Two Reflections Theorem for
Rotations
l
F”
140
°
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70
°
m
F
2.5: Conjectures that Lead to Theorems
Point P is reflected over line m and then
over line n. If the overall result is a
rotation of 80 degrees, what is the
measure of the acute angle formed by
lines m and n?
A. 20
B. 40
C. 80
D. 140
E. 160
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2.5: Conjectures that Lead to Theorems
Assignment
pages 121 – 125,
# 10 – 24 (evens), 25 – 27, 30, 32 – 34, 36, 38,
47, 48, 50
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2.5: Conjectures that Lead to Theorems