2.7 Proving Segment Relationships

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# 2.7 Proving Segment Relationships - PowerPoint PPT Presentation

2.7 Proving Segment Relationships. Objectives. Write proofs involving segment addition Write proofs involving segment congruence. Ruler Postulate. Postulate 2.8 ( Ruler Postulate )

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## 2.7 Proving Segment Relationships

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Presentation Transcript
Objectives
• Write proofs involving segment addition
• Write proofs involving segment congruence
Ruler Postulate

Postulate 2.8 (Ruler Postulate)

The points on any line or line segment can be paired with real numbers so that, given any two points A and B on a line, A corresponds to zero, and B corresponds to a positive real number.

 Basically, all segments have a measure.

Postulate 2.9 (Segment Addition Postulate)

If B is between A and C, then AB + BC = AC.

If AB + BC = AC, then B is between A and C.

AB BC

A B C

AC

Prove the following.

Given: PR = QS

Prove: PQ = RS

Proof:

Statements Reasons

1.

1. Given

PR = QS

2.

2. Subtraction Property

PR – QR = QS – QR

3.

3. Segment Addition Postulate

PR – QR = PQ; QS – QR = RS

4.

4. Substitution

PQ = RS

Example 1:

Proof:

Statements Reasons

1.

1. Given

AC = AB, AB = BX

2.

2. Transitive Property

AC = BX

CY = XD

3.

3. Given

4.

AC + CY = BX + XD

5.

5. Segment Addition Property

AC + CY = AY; BX + XD = BD

6.

6. Substitution

AY = BD

Segment Congruence

Theorem 2.2 (Segment Congruence)

Congruence of segments is reflexive, symmetric, and transitive.

Reflexive Property: AB AB

Symmetric Property: If AB  CD, then CD  AB.

Transitive Property: If AB  CD and CD  EF, then AB  EF.

Proof:

Statements Reasons

1. Given

1.

2. Definition of congruent segments

2.

3.

3. Given

4. Transitive Property

4.

5. Transitive Property

5.

Example 2:

Statements Reasons

1.

1. Given

2.

2. Transitive Property

3.

3. Given

4.

4. Transitive Property

5.

5. Symmetric Property