1. Chapter1: Triangle Midpoint Theorem and Intercept Theorem
Outline
Basic concepts and facts
Proof and presentation
Midpoint Theorem
Intercept Theorem

2. 1.1. Basic concepts and facts
In-Class-Activity 1.
(a) State the definition of the following terms:
Parallel lines,
Congruent triangles,
Similar triangles:

3. 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.
[Figure1]

4. (b) List as many sufficient conditions as possible for
two lines to be parallel,
two triangles to be congruent,
two triangles to be similar

5. 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

6. Corresponding angles
Alternative angles

7. Conditions for two triangles to be congruent
S.A.S
A.S.A
S.S.S

8. Conditions for two triangles similar
Similar to the same triangle
A.A
S.A.S
S.S.S

9. 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.

10. Given the figure in which
To prove that AC=BC.
The plan is to prove that

11. Proof 1. 2. 3. 4. 5. CD=CD 6. 7. AC=BC 1. Given 2. Given 3. By 2
Statements Reasons 1. 2. 3. 4.
CD=CD 6.
7. AC=BC
1. Given
2. Given
3. By 2
4. By 2
5. Same segment
6. A.S.A
7. Corresponding sides
of congruent triangles are equal

12. Example 2 In the triangle ABC, D is an interior point of BC
Example 2 In the triangle ABC, D is an interior point of BC. AF bisects BAD. Show that ABC+ADC=2AFC.

13. Given in Figure BAF=DAF.
To prove ABC+ADC=2AFC.
The plan is to use the properties of angles in a triangle

14. 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)

15. A proof is a sequence of statements, where each statement is either
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.

16. 1.3. Midpoint Theorem
Figure2

17. 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.

18. Given in the figure , AD=CD, BE=CE.
To prove DE// AB and DE=
Plan: to prove ~

19. Proof
Statements
Reasons 1.
2. AC:DC=BC:EC=2 ~ 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 angles
7. By 4 and 2
8. By 7.

20. 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.

21. Example 3 The median of a trapezoid is parallel to the bases and equal to one half of the sum of bases.
Figure
Complete the proof

22. 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

23. 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?

24. 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(?) .
Figure
Write down the complete proof

25. Example 5 In triangle ABC, suppose AE=BF, AC//EK//FJ.
(a) Prove CK=BJ.
(b) Prove EK+FJ=AC.

26. (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

27. 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.

28. Please submit the solutions of (1) In –class-exercise on pg (2) another 4 problems in Tutorial next time THANK YOU
Zhao Dongsheng
MME/NIE
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