Outline: Basic Concepts and Facts Proof and Presentation Midpoint Theorem Intercept Theorem

Outline
•Basic concepts and facts

•Proof and presentation
•Midpoint Theorem
•Intercept Theorem
1.1. Basic concepts and facts
In-Class-Activity
(a) State the definition of the following terms:
Parallel lines,
Congruent triangles,
Similar triangles:
•Two lines are parallel if they do not meet
at any point
•Two triangles are congruent if their

corresponding angles and corresponding
sides equal
•Two triangles are similar if their

Corresponding angles equal and their
corresponding sides are in proportion.
(b) List as many sufficient conditions as possible for
two lines to be parallel,
two triangles to be congruent,
two triangles to be similar

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 transversal ,
(a) two alternative interior (exterior) angles are
equal.
(b) two corresponding angles are equal
(c) two interior angles on the same side of
the transversal are supplement
Corresponding angles
Alternative angles
Conditions for two triangles to be congruent
S.A.S
A.S.A
S.S.S
Conditions for two triangles similar
Similar to the same triangle
A.A
S.A.S
S.S.S
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 ACBand CD
is perpendicular to AB. Prove AC is equal to
CB. C
A D B
C
Given the figure in which
ACD  BCD, CD  AB
To prove that AC=BC.
A D B
The plan is to prove that
ACD  BCD
C
Proof
Statements A D B Reasons
1. ACD  BCD 1. Given

2. CD  AB 2. Given
3. CDA  900 3. By 2
4. CDB  900 4. By 2
5. CD=CD 5. Same segment
6. ACD  BCD 6. A.S.A
7. AC=BC 7. Corresponding sides
of congruent
triangles are equal
Example 2 In the triangle ABC, D is an interior point
of BC. AF bisects BAD. Show that
ABC+ADC=2AFC.
A C
Given in Figure BAF=DAF.
To prove ABC+ADC=2AFC.
The plan is to use the properties of angles in a triangle

Proof: (Another format of presenting a proof)
1. AF is a bisector of BAD,
so BAD=2BAF.
2. AFC=ABC+BAF (Exterior angle )
3. ADC=BAD+ABC (Exterior angle)

=2BAF +ABC (by 1)
4. ADC+ABC
=2BAF +ABC+ ABC ( by 3)
=2BAF +2ABC
=2(BAF +ABC)
=2AFC. (by 2)
What is a proof?
A proof is a sequence of statements, where each
statement is either an assumption, or a statement
derived from the previous statements, or an accepted
statement.
The last statement in the sequence is the
conclusion.
1.3. Midpoint Theorem
C
D E
A B
1.3. Midpoint Theorem
Theorem 1 [ Triangle Midpoint Theorem]

The line segment connecting the midpoints
of two sides of a triangle
is parallel to the third side
and
is half as long as the third side.
Given in the figure , AD=CD, BE=CE.
To prove DE// AB and DE= 1
AB
2
Plan: to prove ~
ACB DCE
D E
A B
Statements Reasons
Proof
1. ACB  DCE 1. Same angle
2. AC:DC=BC:EC=2 2. Given
4. ACB ~ DCE 4. S.A.S
5. CAB  CDE 5. Corresponding
angles of similar
triangles
6. corresponding angles
6. DE // AB
7. DE:AB=DC:CA=2 7. By 4 and 2
8. By 7.
8. DE= 1/2AB
In-Class Activity 2 (Generalization and extension)
If in the midpoint theorem we assume AD and BE are

one quarter of AC and BC respectively, how should we
change the conclusions?
State and prove a general theorem of which the

midpoint theorem is a special case.
Example 3 The median of a trapezoid is parallel to the
bases and equal to one half of the sum of bases.
A B
E F
D C
Complete the proof

Example 4 ( Right triangle median theorem)
The measure of the median on the
hypotenuse of a right triangle is one-half of
the measure of the hypotenuse.
Read the proof on the notes

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?
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(?) . C
Figure
D E
Write down the complete A B

proof
Example 5 In triangle ABC, suppose AE=BF,
AC//EK//FJ.
(a) Prove CK=BJ.
(b) Prove EK+FJ=AC.
C
K
A E F B
(a)
1 KJ EF

2. BJ BF
BK BE

3. BJ BF
AE CK
4. 
BE BK
5. AE BE

CK BK
6. AE BF

CK BJ
7. Ck=BJ
CK AE
 1
BJ BF
(b) Link the mid points of EF and KJ. Then use
the midline theorem for trapezoid
In-Class-Exercise
In ABC, the points D and F are on side AB,
point E is on side AC.
(1) Suppose that
DE // BC , FE // DC , AF  4, FD  6
Draw the figure, then find DB.
( 2 ) Find DB if AF=a and FD=b.

THANK YOU

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Outline

•Basic concepts and facts

•Two triangles are congruent if their

•Two triangles are similar if their

two lines to be parallel,

two triangles to be congruent,

two triangles to be similar

Given the figure in which

The plan is to prove that

1. ACD  BCD 1. Given

The plan is to use the properties of angles in a triangle

3. ADC=BAD+ABC (Exterior angle)

Theorem 1 [ Triangle Midpoint Theorem]

If in the midpoint theorem we assume AD and BE are

State and prove a general theorem of which the

Complete the proof

Read the proof on the notes

Suppose in a triangle the measure of a

Theorem 2 [Triangle Intercept Theorem]

Write down the complete A B

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

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