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

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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

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CB

A

E

D

Statements Reasons

1) 1)

2) 2)

3) 3)

4) 4)

5) 5)

6) 6)

CDA BDE GivenGiven bisects DB CDA

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CDB BDA Def. Angle BisectorGiven

Transitive Postulate (3,5)

bisects DA BDE��������������

Def. Angle BisectorBDA ADE CDB ADE