Chapter1 Triangle Midpoint Theorem and

Intercept Theorem

- Outline
- Basic concepts and facts
- Proof and presentation
- Midpoint Theorem
- Intercept Theorem

1.1. Basic concepts and facts

- In-Class-Activity 1.
- (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. - Figure1

- (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 and CD
- is perpendicular to AB. Prove AC is equal

to CB.

- Given the figure in which
- To prove that ACBC.
- The plan is to prove that

Proof

Statements

Reasons

1. 2. 3. 4. 5. CDCD 6. 7. ACBC 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

- Example 2 In the triangle ABC, D is an

interior point of BC. AF bisects ?BAD. Show

that ?ABC?ADC2?AFC.

- Given in Figure ?BAF?DAF.
- To prove ?ABC?ADC2?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 ?BAD2?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

Figure2

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 , ADCD, BECE.
- To prove DE// AB and DE
- Plan to prove

Proof

Statements Reasons

1. 2. ACDCBCEC2 4. 5. 6. DE // AB 7. DEABDCCA2 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.

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.

Figure

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(?) .

Figure

Write down the complete proof

- Example 5 In triangle ABC, suppose AEBF,

AC//EK//FJ. - (a) Prove CKBJ.
- (b) Prove EKFJAC.

- (a)
- 1
- 2.
- 3.
- 4.
- 5.
- 6.
- 7. CkBJ
- (b) Link the mid points of EF and KJ. Then use
- the midline theorem for trapezoid

- 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 AFa and FDb.

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_at_nie.edu.sg