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Chapter1: Triangle Midpoint Theorem

and Intercept Theorem

Outline

Basic concepts and factsProof and presentation

Midpoint Theorem

Intercept Theorem

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1.1. Basic concepts and facts

In-Class-Activity 1.

(a) State the definition of the following terms:

Parallel lines,

Congruent triangles,

Similar triangles:

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Two lines are parallelif they do not meet

at any point

Two triangles are congruentif their

corresponding angles and correspondingsides equal

Two triangles are similarif their

Corresponding angles equal and theircorresponding sides are in proportion.

[Figure1]

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(b) List as many sufficient conditions as

possible for

two lines to be parallel,

two triangles to be congruent,

two triangles to be similar

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Conditions for lines two be parallel

two lines perpendicular to the same line.

two lines parallel to a third line

If two lines are cut by a t ransversal ,

(a) two alternat ive in ter ior (exter ior) ang lesare

equal.

(b) two corresponding anglesare equal

(c) two in ter ior angles on the same side of

the transversal are supplement

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Corresponding angles

Alternative angles

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Conditions for two triangles to be congruent

S.A.S

A.S.A

S.S.S

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Conditions for two triangles similar

Similar to the same triangle

A.A

S.A.S

S.S.S

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1.2. Proofs and presentation

What is a proof? How to present a proof?

Example 1 Suppose in the figure ,

CD is a bisector of and CD

is perpendicular to AB. Prove AC is equal

to CB.

ACB

DA B

C

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Given the figure in which

To prove that AC=BC.

The plan is to prove that

ABCDBCDACD ,

BCDACD

DA B

C

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Proof

1.

2.

3.4.

5. CD=CD

6.7. AC=BC

1. Given

2. Given

3. By 24. By 2

5. Same segment

6. A.S.A7. Corresponding sides

of congruent

triangles are equal

BCDACD

ABCD

090CDA0

90CDB

BCDACD

Statements ReasonsDA B

C

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Example 2 In the triangle ABC, D is an

interior point of BC. AF bisects

BAD.Show that ABC+ADC=2AFC.

A C

B

D

F

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Given in Figure BAF=DAF.

To prove ABC+ADC=2AFC.

The plan is to use the properties of angles in

a triangle

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Proof: (Another format of presenting a proof)

1. AF is a bisector of BAD,

so BAD=2BAF.2. AFC=ABC+BAF (Exterior angle)

3. ADC=BAD+ABC (Exterior angle)

=2BAF +ABC (by 1)

4. ADC+ABC

=2BAF +ABC+ ABC ( by 3)

=2BAF +2ABC=2(BAF +ABC)

=2AFC. (by 2)

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What is a proof?

A proof is a sequence of statements, where

each statement is either

an assumption,

or a statement derivedfrom the previousstatements ,

or an accepted statement.

The last statement in the sequence is theconclusion.

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1.3. Midpoint Theorem

ED

A B

C

Figure2

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1.3. Midpoint Theorem

Theorem 1[ Triangle Midpoint Theorem]

The line segment connecting the midpoints

of two sides of a triangleis parallel to the third side

and

is half as long as the third side.

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Given in the figure , AD=CD, BE=CE.

To prove DE// AB and DE=

Plan: to prove ~

AB2

1

ACB DCE

ED

A B

C

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Proof

Statements Reasons

1.

2. AC:DC=BC:EC=2

4. ~

5.

6. DE // AB

7. DE:AB=DC:CA=2

8. DE= 1/2AB

1. Same angle

2. Given

4. S.A.S

5. Corresponding

angles of similar

triangles

6. corresponding angles7. By 4 and 2

8. By 7.

DCEACB

ACB DCE

CDECAB

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Example3 The median of a trapezoid is

parallel to the bases and equal to one halfof the sum of bases.

FE

CD

A B

Complete the proof

Figure

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Example 4 ( Right triangle median theorem)

The measure of the median on thehypotenuse of a right triangle is one-half of

the measure of the hypotenuse.

E

A

C

B

Read the proof on the notes

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In-Class-Activity 4

(posing the converse problem)

Suppose in a triangle the measure of a

median on a side is one-half of the measure

of that side. Is the triangle a right

triangle?

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1.4 Triangle Intercept Theorem

Theorem 2[Triangle Intercept Theorem]

If a line is parallel to one side of a triangle

it divides the other two sides proportionally.Also converse(?) .

B

C

D E

A

Figure

Write down the complete

proof

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Example 5 IntriangleABC, suppose

AE=BF, AC//EK//FJ.(a) Prove CK=BJ.

(b) Prove EK+FJ=AC.

J

K

A

C

BE F

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(a)

1

2.

3.

4.5.

6.

7. Ck=BJ(b) Link the mid points of EF and KJ. Then use

the midline theorem for trapezoid

BF

EF

BJ

KJ

BF

BE

BJ

BK

BK

CK

BE

AE

BK

BE

CK

AE

BJ

BF

CK

AE

1BF

AE

BJ

CK

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In-Class-Exercise

In , the points D and F are on side AB,

point E is on side AC.

(1) Suppose that

Draw the figure, then find DB.

( 2 ) Find DB if AF=a and FD=b.

ABC

6,4,//,// FDAFDCFEBCDE

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Please submit the solutions of

(1) In class-exercise on pg 7(2) another 4 problems in

Tutorial 1

next time.

THANK YOU

Zhao Dongsheng

MME/NIE

Tel: 67903893

E-mail: dszhao@nie.edu.sg