GeometryGeometry

2.3 Proving Theorems2.3 Proving Theorems

IntroIntro

Theorems are statements that are Theorems are statements that are proved.proved.

They are deduced from postulates, They are deduced from postulates,

statements that are accepted without proof.statements that are accepted without proof.

POE’s are treated as postulatesPOE’s are treated as postulates

Deductive ReasoningDeductive Reasoning uses postulates uses postulates defn.’s, thm.’s, and given information.defn.’s, thm.’s, and given information.

Hint: deductive = definition

4 Reasons Used in Proofs4 Reasons Used in Proofs

Given InformationGiven Information

Defn.’sDefn.’s

Postulates(POE’s)Postulates(POE’s)

Theorems from yesterday i.e.(that is) Theorems from yesterday i.e.(that is) theorems that have been proventheorems that have been proven

Midpoint Thm.Midpoint Thm.

If M is the midpoint of , If M is the midpoint of , then AM = (1/2)AB and MB = then AM = (1/2)AB and MB = (1/2)AB(1/2)AB

How is this different from the midpoint defn.?How is this different from the midpoint defn.? Key: The midpoint theorem uses ½. Key: The midpoint theorem uses ½.

AB

A

.M

.B

.

StatementsStatements1)1) M is the midpoint ofM is the midpoint of

2)2) AM = MBAM = MB

3)3) AM + MB = ABAM + MB = AB

4)4) AM + AM = AB or 2AM = AM + AM = AB or 2AM = ABAB

5)5) AM = (1/2)ABAM = (1/2)AB

6)6) MB = (1/2)ABMB = (1/2)AB

Now that the Midpoint Thm. has been Now that the Midpoint Thm. has been proven, it may be used as a reason proven, it may be used as a reason in a proof!in a proof!

ReasonsReasons1)1) GivenGiven

2)2) Defn. of MidpointDefn. of Midpoint

3)3) Segment Add. Post.Segment Add. Post.

4)4) Substitution Substitution (Steps 2 & 3)(Steps 2 & 3)

5)5) Division POEDivision POE

6)6) Substitution Substitution (Steps 2 & 5)(Steps 2 & 5)

Proof of the Midpoint TheoremProof of the Midpoint Theorem

G: M is the midpoint ofG: M is the midpoint of

P: AM = (1/2)AB; MB = (1/2)ABP: AM = (1/2)AB; MB = (1/2)AB

AB

A

.M

.B

.AB

Midpoint Defn. versus Midpoint Thm.Midpoint Defn. versus Midpoint Thm. (uses ½)(uses ½)

If Y is the midpoint of , If Y is the midpoint of ,

……then what is true by the reason of then what is true by the reason of midpoint defn.?midpoint defn.?

Answer: AY = YBAnswer: AY = YB

……then what is true by the reason of then what is true by the reason of midpoint thm.?midpoint thm.?

Answer: AY = (1/2)AB or Answer: AY = (1/2)AB or YB = (1/2)ABYB = (1/2)AB

AB.Y

.A

.B

Angle Bisector Them.Angle Bisector Them.

If BX is the bisector of , then If BX is the bisector of , then

How is this different from the angle bisector defn.?How is this different from the angle bisector defn.?

Key: The theorem uses ½.Key: The theorem uses ½.

ABC

ABCmXBCmABCmABXm 2

1 and

2

1

.C

B .

. X. A

Angle Bisector Thm. Versus the Angle Angle Bisector Thm. Versus the Angle Bisector Defn.Bisector Defn.

If BX is the bisector of , thenIf BX is the bisector of , then

is true by the reason of __________? is true by the reason of __________?

Answer: Angle Bisector Defn.Answer: Angle Bisector Defn. If BX is the bisector of , thenIf BX is the bisector of , then

is true by the reason of __________? is true by the reason of __________?

Answer:Answer: Angle Bisector Thm.Angle Bisector Thm.

ABC

ABCmXBCmABCmABXm 2

1 and

2

1

.C

B.

. X. A

XBCmABXm

ABC

Please turn your books to P. Please turn your books to P. 4545

1)1) Angle Add. Post.Angle Add. Post.

2)2) Segment Add. Post.Segment Add. Post.

3)3) Angle Add. Post.Angle Add. Post.

4)4) Midpoint Defn.Midpoint Defn.

5)5) Midpoint Thm.Midpoint Thm.

6)6) Segment Bisector Segment Bisector Defn.Defn.

7)7) Segment Bisector Segment Bisector Defn.Defn.

8)8) Angle Bisector Thm.Angle Bisector Thm.

9)9) Angle Bisector Defn.Angle Bisector Defn.

10) Reasons10) Reasons

1)1) GivenGiven

2)2) m<XBC or <XBC by m<XBC or <XBC by Angle Bisector Defn.Angle Bisector Defn.

3)3) Angle Add. Post.Angle Add. Post.4)4) Substitution Substitution (Steps 2 & (Steps 2 &

3)3)

5)5) Mult. POEMult. POE6)6) SubstitutionSubstitution (Steps 2 & (Steps 2 &

5)5)

HWHW

P. 41 #4-12 (4X)P. 41 #4-12 (4X)

P.46 #1-19 OddP.46 #1-19 Odd

P. 51 CE #1-21 Odd P. 51 CE #1-21 Odd

Quiz 2.1-2.3 on WednesdayQuiz 2.1-2.3 on Wednesday