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Geometry
Chapter 2: Reasoning and Proof
2.1 Inductive objectives:
-make conjectures based on inductive reasoning
Reasoning
-find counterexamples
and
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Conjecture
Answer question on transparency. Show work here and circle your final answer:
Bell Ringer:
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conjecture
inductive
reasoning
counterexample
Make a conjecture about the next item in the sequence.
Examples
Text Page 64
Determine whether each conjecture is true or false. Give a counterexample for any false
conjecture.
Assignment 2.1
page 64
#12- 36 even
(omit#18),48,53,
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59
Summary: How do you prove that a conjecture is false? Provide one example.
Honors:
Additionally,
#38-40, 43
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2.3
Conditional
Statements
Bell Ringer
objective:
-analyze statements in if-then form
-write the converse, inverse, and contrapositive of if-then statements
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Refer to the transparency (show your work and circle your answer:
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conditional
statement
hypothesis
conclusion
Examples
Write each statement in the if-then form. Identify the hypothesis and conclusion.
a. A 32-ounce pitcher holds a quart of liquid
b. The sum of the measures of supplementary angles is 180.
c. An angle formed by perpendicular lines is a right angle
Other Types of
statements
Summary:
Statement
Formed by
Given hypothesis
and conclusion
Exchanging the
Converse
hyp. and concl.of
the conditional
Negating both the
Inverse
hypo. and concl. of
the conditional
Contrapositive Negating both the
hyp. and concl. of
the converse
Conditional
Examples
Truth
Value
If an animal is a dog,
then it is a mammal
If an animal is a
mammal, then it is a
dog.
If an animal is not a
dog, then it is not a
mammal
If an animal is not a
mammal, then it is not
a dog
Assignment 2.3
page 78 #18-44
even, 61-62
Honors:
Additionally,
46,47,50
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2.4 Deductive objective:
Reasoning -use deductive reasoning to draw conclusions
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Bell Ringer
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Deductive
reasoning
the process of using facts, rules, definitions, or properties to reach conclusions.
Law of If p q is true and p is true, then q is true.
Detachment
Determine whether each conclusion is valid based on the true conditional given. If not,
Examples write invalid. Explain your reasoning.
If two angles are complementary to the same angle, then the angles are congruent.
1. Given: A and C are complementary to B.
Conclusion: A is congruent to C.
2. Given: A  C
Conclusion: A and C are complements of B.
3. Given: E and F are complementary to G.
Conclusion: E and F are vertical angles.
Law of If p q is true and q r is true, then p r is also true.
Syllogism
Determine whether you can reach a valid conclusion from each set of statements.
Examples 1. If a dog eats Superdog Dog Food, he will be happy.
Rover is happy.
2. If an angle is supplementary to an obtuse angle, then it is acute.
If an angle is acute, then its measure is less than 90.
3. If the measure of A is less than 90, then A is acute.
If A is acute, then A B.
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Summary: What did I learn today? (2 or more sentences)
Assignment 2.4
Pg 85 12-26 even
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2.5
Postulates &
Paragraph
Proofs
Objective:
-identify and use basic postulates about points, lines and planes
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Bell Ringer:
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Postulate a statement that is accepted as true.
(axiom)
Postulate 2.1: Through any two points, there is exactly one line.
Postulate 2.2: Through any three points not on the same line, there is exactly one plane.
Postulate 2.3: A line contains at least two points.
Postulate 2.4: A plane contains at least three points not on the same line.
Postulate 2.5: If two points lie in a plane, then the line containing those points lies in the plane.
Postulate 2.6: If two lines intersect, then their intersection is exactly one point.
Postulate 2.7: If two planes intersect, then their intersection is a line.
Examples
Theorem A statement that can be proved true
Midpoint
Theorem If M is the midpoint of AB , then AM  MB.
Summary: Describe the difference between a postulate and a theorem.
Assignment 2.5
page 92 #16-27
Honors: #28 too
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2.6.
Algebraic
Proofs
Objective:
-Use algebra to write two-column proofs
Bell Ringer (To be collected): Please show your work on a sheet of looseleaf.
Reflexive
For every number a, a = a.
property
Symmetric
For all numbers a and b, if a = b then b = a.
Property
Transitive
For all numbers a, b, and c, if a= b and b = c then a = c.
property
Addition &
For all numbers a, b, and c, if a = b then a + c = b + c and a - c = b - c.
Subtraction
Property
Multiplication /
For all numbers a, b, and c, if a = b then a • c = b• c, and if c ≠ 0 then
Division
a/c = b/c.
Property
Substitution
For all numbers a and b, if a = b then a may be replaced by b in any equation
Property
or expression.
Distributive
Property
Reflexive
Property
For all numbers a, b, and c, a(b + c) = ab + ac.
Segments
AB = AB
Angles
m1 = m1
Symmetric
Property If AB = CD, then CD = AB
If m1 = m2, then m2 = m1
Transitive If AB = CD and CD = EF,
Property then AB = EF
If m1 = m2 and m2 = m3,
then m1 = m3
Two-Column Contains statements and reasons organized in two columns. Each step is called a statement and
Proof the properties, definitions, postulates, or theorems that justify each step are called reasons.
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Examples
Complete each proof.
Assignment 2.6
Part 1, page 97
#14-23 all
Assignment 2.6
Part 2, page 98
#24-27, 37, 38
Summary (below): Describe a two column proof in your own words.
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Objectives:
2.7 Proving
Segment
Relationships _____________________________________________________________________________
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Bell Ringer
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Segment
addition B is between A and C if and only if AB+ BC = AC.
Postulate
Congruence of segments is reflexive, symmetric, and transitive.
AB  AB
Symmetric Property:
If AB  CD, then CD  AB.
Transitive Property:
If AB  CD and CD  EF, then AB  EF.
Theorem 2.2 Reflexive Property:
Examples
(page 103 #4-7)
Assignment 2.7
page 104 #12-21
all, 29, 30
Honors:
Additionally,
22,23, & 26
Summary (on back of notes): Describe the theorems that relate to congruence of segments.
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Objective:
2.8 Proving
-Write proofs involving supplementary and complementary angles
Angle
Relationships -Write proofs involving congruent and right angles
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Bell Ringer:
Refer to the transparency. Show work here and circle your answer:
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Angle Addition R is in the interior of PQS if and only if
Postulate
mPQR + mRQS = mPQS.
Supplement If two angles form a linear pair, then they are supplementary angles.
If 1 and 2 form a linear pair, then m1 + m2 = 180.
Theorem
Complement If the non-common sides of two adjacent angles form a right angle,
Theorem then the angles are complementary angles.
If GF  GH, then m3 + m4 = 90.
Vertical Angle If two angles are vertical angles, then they are congruent.
Theorem
m6  m7
Examples
(See Wkst 2.8
Skills Practice)
Assignment
page 112 #1624, 27-32, 38, 39
Summary: What did I learn today? What do I need help with?
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