Aim: What are the first postulates used in
geometry proofs?
Do Now:
1) You stand in front of a full-length mirror.
How tall is your reflection?
Ans.: Same height as myself
2) John is the same height as Lisa. What
conclusion can you make?
Ans.: Lisa is the same height as John.
3)Frank is the same age as Javier. Javier is
the same age as Patricia. What conclusion
can you make?
Ans.: Patricia is the same age as Frank.
Geometry Lesson: First Postulates
1
2-Column Geometry Proofs
Given: Information “given” to help start the proof.
Prove: The final conclusion that we must make based
on the given, postulates, definitions and theorems.
Arrange proof in 2-column table format. Every
statement must have an accompanying reason.
Statements
1)Given usually goes first.
2)Conclusions…
3)Conclusions…
4)Prove is last statement
Reasons
1) Given…
2) Definitions, Postulates, Theorems…
3) Definitions, Postulates, Theorems…
4) Definitions, Postulates, Theorems…
Geometry Lesson: First Postulates
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Postulates and Theorems
Postulate: A postulate is a statement that we accept
as true without proof.
Theorem: A theorem is a statement that can be
proved by deductive reasoning.
Geometry Lesson: First Postulates
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First Postulates – a.k.a. Properties of Equality
Postulate #1 – Reflexive Property of Equality:
“A quantity is equal to itself.” a = a
“A line segment (or angle) is congruent to itself.” a  a
Ex 1:
Ex 2:
B
C
D
A
CD  CD
D
B
Reflexive
Postulate
A
ABC  ABC
Geometry Lesson: First Postulates
E
C
Reflexive
Postulate
4
Postulate #2 - Symmetric Property of Equality:
“An equality may be expressed in either order.”
If a = b,
If a  b
then, b = a
T hen, b  a
Ex 1:
Ex 2:
Given: A B  C D
A
B
Given:  P   Q
C
CD  AB
P
Q
D
Symmetric
Postulate
Q  P
Geometry Lesson: First Postulates
Symmetric
Postulate
5
Postulate #3 – Transitive Property of Equality:
If quantities are equal to the same quantity, then they
are equal to each other. If a = b
If
and b = c,
then a = c
C
G iven: A B  B C , B C  C A
P rove:
A B C is equilateral
Statements
A
a b
an d b  c ,
th en a  c .
B
Reasons
1) A B  B C
1) Given
2) B C  C A
2) Given
3) A B  C A
3) Transitive Postulate
Def.
triangle
4) A B C is equilateralGeometry4)Lesson:
First equilateral
Postulates
6
Given:
mx
Example:
Prove:
 40, m  y  40
x  y
x
Statements
1)
2)
3)
4)
5)
m  x  40
m  y  40
40  m  y
mx  my
x  y
y
Reasons
1)
2)
3)
4)
5)
Given
Given
Symmetric Postulate
Transitive Postulate
Def. congruent angles
Geometry Lesson: First Postulates
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Examples: First Postulates
1)
G iven: A B  LM , C D  R S , LM  R S
P rove: A B  C D
2)
A
B
C
D
L
M
R
S
G iven:  C D A   B D E ,
C
B
D B bisects  C D A ,
D A bisects  B D E
A
D
P rove:  C D B   A D E
Geometry Lesson: First Postulates
E
8
GEx
iven:
#1: A B  LM , C D  R S , LM  R S
P rove: A B  C D
A
B
C
D
L
M
R
S
Statements
1)
2)
3)
4)
5)
6)
7)
AB  LM
LM  R S
AB  RS
CD  RS
RS  CD
AB  CD
AB  CD
Reasons
1)
2)
3)
4)
5)
6)
7)
Given
Given
Transitive Postulate
Given
Symmetric Postulate
Transitive Postulate
Def. congruent line segments
Geometry Lesson: First Postulates
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G iven:  C D A   B D E ,
C
Ex#2
B
D B bisects  C D A ,
D A bisects  B D E
A
D
P rove:  C D B   A D E
E
Statements
1)  C D A   B D E
2) D B bisects  C D A
3)  C D B   B D A
4) D A bisects  B D E
5)  B D A   A D E
6)  C D B   A D E
Reasons
1)
2)
3)
4)
5)
6)
Given
Given
Def. Angle Bisector
Given
Def. Angle Bisector
Transitive Postulate (3,5)
Geometry Lesson: First Postulates
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