Geometry

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Geometry

• Chapter 2 Logic and Reasoning

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2.5 Proofs and Segments

• Objectives
• Day 1 Set up the given, prove and picture for
  deductive reasoning
• Day 2 Justify statements in a proof

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What you know

• Properties
• Definition of Congruence
• Definition of Midpoints
• Segment Addition

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Setting up a Geometric proof

• Step 1 Write the statement in if then form
• Vertical Angles are congruent
• If ______________ then ____________

Angles are vertical
they are congruent
Angles are vertical
they are congruent
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Setting up a Geometric proof

• Step 2 Draw the given and label the parts

Angles are vertical
they are congruent
Picture
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Setting up a Geometric proof

• Step 3 Rewrite the given using labels and
  symbols from the picture

Angles are vertical
they are congruent
?AEC,
?DEB
are vertical
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Setting up a Geometric proof

• Step 4 Rewrite the conjecture using labels and
  symbols from the picture

Angles are vertical
they are congruent
?AEC,
?DEB
are vertical
?AEC ? ?DEB
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Example
A square has four right angles
it has four right angles
it is a square
it has four right angles
it is a square
?DAC, ?ABC, ?BCD, ?CDA are right angles
ABCD is a square
A
B
C
D
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Two segments that have equal measures are
congruent
they are congruent
two segments that have equal measures
two segments that have equal measures
they are congruent
AB CD

?
AB CD
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Day 2 Vocabulary Other Properties
Definition of Congruence
If m?1m?2 Then ?1??2
If AB ? CD Then AB CD
A
Angle Addition Property
m?ABCm?CBDm?ABD
C
B
D
E
D
F
DEEF DF
Segment Addition Property
If M is the midpoint of PQ Then PM ? MQ
Definition of Midpoint
M
P
Q
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Proof Tricks

• Substitutions

AB BC (Given)
AB DE (substitution)
DE BC (Given)
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Proof Tricks

• Substitution

ADDBAB (Segment Addition)
AB BC (Given)
ADDBBEEC (Substitution)
BEECBC (Segment Addition)
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Proof Tricks

• Subtraction

ABBCBCDE (Given)
ABDE (Subtraction P.O.E)
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Brainstorming ideas for a proof Doing a proof
Given NL NM AL BM Prove NA
NB
AL NA NL
BM NB NM
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Work
AL NA NL (Angle Addition)
Given NL NM AL BM
AL NA BM NB (Substitution)
AL NA AL NB (Substitution)
BM NB NM (Angle Addition)
NA NB (Subtraction)
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Topic 3 Example
Given NL NM AL BM Prove NA
NB
Brainstorm Little Little Big
Midpoint
Then these are the same
AL NA NL
If theses are the same
BM NB NM
Tricks to try
AL NA BM NB
BM NA NB BM
-BM -BM
We want to keep these
NA NB
IF we replace AL with BM we could subtract BM
from both sides
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Topic 4 Example Cont.
M
L
Given NL NM AL BM Prove NA
NB
B
A
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Practice

• Given m?1m?2
• Prove m?PXRm?SXQ

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Assignment

• Day 1
• Set up a proof for each of the following
  statements
• The diagonals of a square are congruent
• Two perpendicular lines form four right angles.
• Day 2
• Page 96 26,
• Page 103 12, 34