Geometry
2.3 Proving Theorems
Intro

Theorems are statements that are proved.

They are deduced from postulates, statements

POE’s are treated as postulates

Deductive Reasoning uses postulates
defn.’s, thm.’s, and given information.
that are accepted without proof.
Hint: deductive = definition
4 Reasons Used in Proofs

Given Information

Defn.’s

Postulates(POE’s)

Theorems from yesterday i.e.(that is)
theorems that have been proven
Midpoint Thm.

If M is the midpoint of AB,
then AM = (1/2)AB and MB = (1/2)AB
A
.
M
.
B
.
How is this different from the midpoint defn.?
Key: The midpoint theorem uses ½.
Proof of the Midpoint Theorem
G: M is the midpoint of AB
P: AM = (1/2)AB; MB = (1/2)AB
Statements
1)
M is the midpoint of
AB
A
M
.
.
Reasons
1) Given
2)
AM = MB
2)
Defn. of Midpoint
3)
AM + MB = AB
3)
Segment Add. Post.
4)
AM + AM = AB or 2AM = AB
4)
Substitution
5)
AM = (1/2)AB
5)
Division POE
6)
MB = (1/2)AB
6)
Substitution
Now that the Midpoint Thm. has been proven, it
may be used as a reason in a proof!
(Steps 2 & 3)
(Steps 2 & 5)
B
.
Midpoint Defn. versus Midpoint Thm.
.

If Y is the midpoint of AB ,
A
.
Y
…then what is true by the reason of
midpoint defn.?
Answer: AY = YB
…then what is true by the reason of
midpoint thm.?
Answer: AY = (1/2)AB or
YB = (1/2)AB
(uses ½)
.
B
Angle Bisector Them.

If BX is the bisector of
ABC , then
1
1
mABX  mABC and mXBC  mABC
2
2
.
A
.
X
.
B
.
C
How is this different from the angle bisector defn.?
Key: The theorem uses ½.
Angle Bisector Thm. Versus the Angle Bisector
Defn.
If BX is the bisector of ABC , then
mABX  mXBC is true by the reason of __________?

Answer: Angle Bisector Defn.
 If BX is the bisector of ABC , then
1
1
mABX  mABC and mXBC  mABC
2
2
. A
is true by the reason of __________?
. X
.
B.
C
Answer: Angle Bisector Thm.
Please turn your books to P. 45
1)
2)
3)
4)
5)
6)
7)
8)
9)
Angle Add. Post.
Segment Add. Post.
Angle Add. Post.
Midpoint Defn.
Midpoint Thm.
Segment Bisector Defn.
Segment Bisector Defn.
Angle Bisector Thm.
Angle Bisector Defn.
10) Reasons
1) Given
2) m<XBC or <XBC by
Angle Bisector Defn.
3) Angle Add. Post.
4) Substitution (Steps 2 & 3)
5) Mult. POE
6) Substitution (Steps 2 & 5)
HW
P. 41 #4-12 (4X)
P.46 #1-19 Odd
P. 51 CE #1-21 Odd
Quiz 2.1-2.3 on Wednesday
