Transcript Notes Section 2.5 and 2.7

Geometry Notes
Sections 2-5 & 2-7
What you’ll learn
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How to identify and use basic
postulates/axioms and theorems about
points, lines, and planes
Vocabulary
Postulate
 Axiom
 Theorem
 Proof
 Paragraph proof
 Informal proof
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Recall the definition of Postulate
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DEFN Postulate (axiom is another word
for postulate):
 A statement
that describes a fundamental
relationship between the basic terms of
geometry
 Always accepted as true
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DEFN Theorem:
 A statement
or conjecture that can be proven
true using postulates, definitions, and
undefined terms
 Must be proven
Postulates
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Through any two points there is exactly one line.
In conditional format:
If you have two points there is exactly one line
that would go through those two points.
Symbolically:
2 pts →exactly one line
Through any three noncollinear points there is
exactly one plane.
In conditional format:
If you have three noncollinear points, there is
exactly one plane that would contain them.
Symbolically:
3 noncollinear pts →exactly one plane
More Postulates
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If you have a line, then that line has at least two
points on it.
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If you have a plane, then it has at least 3
noncollinear points.
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If 2 points lie in a plane, then the entire line
containing those 2 points lies in that plane.
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If 2 lines intersect, then their intersection is
exactly one point.
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If 2 planes intersect, then their intersection is a
line.
Even
Postulates
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RulerMore
Postulate:
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This postulate guarantees all line segments
have length or measure
 If
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you have AB then mAB or AB exists
Symbolically:
AB → mAB or AB
Segment Addition Postulate (hey we already
know this . . .right?)
If B is between A and C, then AB + BC = AC
And if AB + BC = AC then B is between A and C.
Symbolically:
B is between A and C↔ AB + BC = AC
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Definitions
we know. . .
DEFN:
right angle
 An angle is a right angle iff it measures 90
 Symbolically:
 Right angle ↔90
DEFN: congruent segments
 Segments are congruent iff they have the same measure
 Symbolically:
  ↔=
DEFN: congruent angles
 Angles are congruent iff they have the same measure
 Symbolically:
  ↔=
Postulates (axioms), definitions, and
already proven theorems are the facts and
rules we use to justify our argument in
deductive reasoning.
 Proofs are like puzzles or games.You have
to memorize the postulates, definitions
and theorems—they are the rules to the
game.
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The 5 Essential Parts of a Good Proof
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1. State the theorem or conjecture to be proven.
 Now for the parts we really don’t skip--ever
 Okay, I’m not going to lie, sometimes we skip
2. A list of the given information
 Usually
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is the part that looks like a picture
4. State what is to be proved
 Again,
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cleverly hidden by the word “Given”
3. A diagram of what we’re given (and only what
we are given)
 This
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this one
cleverly hidden by the word “Prove”
5. A system of deductive reasoning
 My
favorite is a toss up between the flow chart proof
and the two – column proof
Things everyone needs to know about writing
proofs:
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The given and prove statements cannot be
written in a general format, they must be specific
Example:
2
angles are right angles is too general
 A and B are right angles is what you want
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The statements and reasons must be numbered
in any proof
You are only allowed to use the word “given”,
postulates, definitions, or previously proven
theorems for reasons
Let’s try one. . .
 Yes a proof.
 Prove that all right angles are congruent.
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Hint: Rewrite the statement you are
proving as a conditional statement (in Ifthen form)
 If two angles are right angles, then they
are congruent.
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 This
is the part we usually skim over, but
since this is our first time we might want to do
all the steps. . .
If two angles are right angles then they are
congruent.
 So, do you think it’s true?
 Why?
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Now that we believe, let’s move on to step
2. . . What are we given to use?
 The
given information is always listed in the
hypothesis of the conditional statement.
 The “If” part
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We are given two right angles. I would feel so
much better if we gave them names. . . It would
make the whole thing more personal.
Let’s call them 1 and 2 (see we can use
numbers sometimes)
Now what did that if part say. . .
 If two angles were right angles. . . .
 Given: 1 and 2 are right angles
Two essential parts covered, three to go.
 What’s next?
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Next is a diagram of our given information
 Given: 1 and 2 are right angles
 We have to draw 2 basic right angles and
name them 1 and 2 – never add special
circumstances like making the angles
adjacent, linear pairs, vertical angles. . .
1
2
Now what do we have so far?
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Given: 1 & 2 are right angles
1
What’s the next step in our list?
Step 4 out of 5. . . 
4. State what is to be proved.
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The information to be proved is found in the
conclusion of the conditional statement
 The
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part after the word “then”
If two angles are right angles, then they are
congruent.
2
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Given: 1 & 2 are right angles
1
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2
Prove: 1  2
Remembe
r they
have
names
And now for the last step
now
5. A system of deductive reasoning
 My
favorite is a toss up between the flow chart proof
and the two – column proof
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Given: 1 & 2 are right angles
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Prove: 1  2
Statements
1
Reasons
1. 1 & 2 are right angles 1. Given
2. m 1 =90 m2=90
2. right s ↔ 90
3. m1 = m2
3. Substitution
4. 1 2
4. = ↔ 
2
Have you learned. . .
How to identify and use basic postulates
about points, lines, and planes?
 We will build on the process of writing
proofs. It takes time. You’ll get there. 
 Assignment : Worksheet 2.7
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