1geometry lesson: first postulates aim: do now: 1) you stand in front of a full-length mirror. how...
TRANSCRIPT
1Geometry Lesson: First Postulates
Aim:
Do Now:
1) You stand in front of a full-length mirror. How tall is your reflection?
2) John is the same height as Lisa. What conclusion can you make?
3)Frank is the same age as Javier. Javier is the same age as Patricia. What conclusion can you make?
Ans.: Same height as myself
Ans.: Lisa is the same height as John.
Ans.: Patricia is the same age as Frank.
What are the first postulates used in geometry proofs?
2Geometry Lesson: First Postulates
2-Column Geometry ProofsGiven: Information “given” to help start the proof.
Prove: The final conclusion that we must make based on the given, postulates, definitions and theorems.
Statements Reasons
1)Given usually goes first. 1) Given…
2)Conclusions… 2) Definitions, Postulates, Theorems…
3)Conclusions… 3) Definitions, Postulates, Theorems…
4)Prove is last statement 4) Definitions, Postulates, Theorems…
Arrange proof in 2-column table format. Every statement must have an accompanying reason.
3Geometry Lesson: First Postulates
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.
4Geometry Lesson: First Postulates
First Postulates – a.k.a. Properties of Equality
Postulate #1 – Reflexive Property of Equality:
“A quantity is equal to itself.” a = a
A B
C
D
Ex 1:
“A line segment (or angle) is congruent to itself.”
a a
A
B
C
ED
Ex 2:
CD CD Reflexive Postulate
ABC ABC Reflexive Postulate
5Geometry Lesson: First Postulates
Postulate #2 - Symmetric Property of Equality:“An equality may be expressed in either order.”
If a = b,
then, b = a
Ex 1:
Given:
A B
C D
AB CD
CD AB Symmetric Postulate
If
Then,
a b
b a
Ex 2:
Given:
P Q
P Q
Q P Symmetric Postulate
6Geometry Lesson: First Postulates
Postulate #3 – Transitive Property of Equality:If quantities are equal to the same quantity, then they are equal to each other. If a = b
and b = c,
then a = c
If
and ,
then .
a b
b c
a c
Given: ,
Prove: is equilateral
AB BC BC CA
ABC
A B
C
Statements Reasons
1) 1)
2) 2)
3) 3)
4) 4)
AB BCBC CAAB CA
is equilateralABC
GivenGiven
Transitive Postulate
Def. equilateral triangle
7Geometry Lesson: First Postulates
Example:Given:
Prove:
40, 40m x m y
x y x y
Statements Reasons
1) 1)
2) 2)
3) 3)
4) 4)
5) 5)
40m x 40m y
40 m y m x m y
x y
Given
Given
Symmetric Postulate
Transitive Postulate
Def. congruent angles
8Geometry Lesson: First Postulates
Given: , ,
Prove:
AB LM CD RS LM RS
AB CD
C D
SRML
A B
Given: ,
bisects ,
bisects
Prove:
CDA BDE
DB CDA
DA BDE
CDB ADE
��������������
��������������
CB
A
E
D
Examples: First Postulates
1)
2)
9Geometry Lesson: First Postulates
Ex #1:
Statements Reasons
1) 1)
2) 2)
3) 3)
4) 4)
5) 5)
6) 6)
7) 7)
AB LM
Given: , ,
Prove:
AB LM CD RS LM RS
AB CD
C D
SRML
A B
Given
LM RS Given
AB RS Transitive Postulate
CD RS GivenRS CD Symmetric Postulate
AB CD Transitive Postulate
AB CD Def. congruent line segments
10Geometry Lesson: First Postulates
Ex#2Given: ,
bisects ,
bisects
Prove:
CDA BDE
DB CDA
DA BDE
CDB ADE
��������������
��������������
CB
A
E
D
Statements Reasons
1) 1)
2) 2)
3) 3)
4) 4)
5) 5)
6) 6)
CDA BDE GivenGiven bisects DB CDA
��������������
CDB BDA Def. Angle BisectorGiven
Transitive Postulate (3,5)
bisects DA BDE��������������
Def. Angle BisectorBDA ADE CDB ADE