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Segment Addition Two basic postulates for working with segments and lengths are the Ruler Postulate, which establishes number lines, and the Segment Addition Postulate, which describes what it means for one point to be between two other points.

Write a two-column proof.

Given: Q is the midpoint of −− PR .

R is the midpoint of −−−

QS .Prove: PR = QSProof:

PQ

RS

A BC

DE P Q

R S

ExercisesComplete each proof.

1. Given: BC = DE Prove: AB + DE = AC

Statements Reasons 1. BC = DE 1. 2. 2. Seg. Add. Post. 3. AB + DE = AC 3.

2. Given: Q is between P and R, R is between Q and S, PR = QS.Prove: PQ = RS

Statements Reasons1. Q is between 1. Given

P and R. 2. PQ + QR = PR 2. 3. R is between 3.

Q and S.4. 4. Seg. Add. Post.5. PR = QS 5. 6. PQ + QR = 6.

QR + RS7. PQ + QR - QR = 7.

QR + RS - QR 8. 8. Substitution

Example

Statements Reasons1. Q is the midpoint of −−

PR .2. PQ = QR3. R is the midpoint of

−−− QS .

4. QR = RS5. PQ + QR = QR + RS

6. PQ = RS

7. PQ + QR = PR, QR + RS = QS8. PR = QS

1. Given2. Definition of midpoint3. Given4. Definition of midpoint5. Addition Property

6. Transitive Property

7. Segment Addition Postulate8. Substitution

Ruler PostulateThe points on any line or line segment can be put into one-to-one correspondence with real numbers.

Segment Addition Postulate

If A, B, and C are collinear, then point B is between A and C if and only if AB + BC = AC.

Proof:Proof:

Study Guide and InterventionProving Segment Relationships

2-7

Chapter 2 43 Glencoe Geometry

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Chapter 2 44 Glencoe Geometry

Segment Congruence Remember that segment measures are reflexive, symmetric, and transitive. Since segments with the same measure are congruent, congruent segments are also reflexive, symmetric, and transitive.

Write a two-column proof.

Given: −− AB " −−−

DE ; −−−

BC " −− EF

Prove: −− AC " −−−

DF Proof:

ExercisesJustify each statement with a property of congruence.

1. If −−− DE "

−−− GH , then

−−− GH " −−−

DE . 2. If −−

AB " −−

RS and −−

RS " −−− WY then

−− AB " −−−

WY . 3. −−

RS " −−

RS

4. Complete the proof. Given: −−

PR " −−−

QS Prove: −−−

PQ " −−

RS Proof:

DE FA

B C

P QR S

Reflexive Property −−

AB " −−

AB

Symmetric Property If −−

AB " −−−

CD , then −−−

CD " −−

AB .

Transitive Property If −−

AB " −−−

CD and −−−

CD " −−

EF , then −−

AB " −−

EF .

Example

Statements Reasons1. −−

AB " −−− DE

2. AB = DE3. −−−

BC " −− EF

4. BC = EF5. AB + BC = DE + EF

6. AB + BC = AC, DE + EF = DF

7. AC = DF8. −−

AC " −−− DF

1. Given2. Definition of congruence of segments3. Given4. Definition of congruence of segments5. Addition Property

6. Segment Addition Postulate

7. Substitution8. Definition of congruence of segments

Statements Reasons

a. −− PR "

−−− QS

b. PR = QSc. PQ + QR = PR d.

e. PQ + QR = QR + RSf. g.

a. b. c. d. Segment Addition Postulatee.

f. Subtraction Property

g. Definition of congruence of segments

Study Guide and Intervention (continued)

Proving Segment Relationships

2-7

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Chapter 2 45 Glencoe Geometry

Justify each statement with a property of equality, a property of congruence, or a postulate.

1. QA = QA

2. If −− AB "

−−− BC and

−−− BC "

−−− CE then

−− AB "

−−− CE .

3. If Q is between P and R, then PR = PQ + QR.

4. If AB + BC = EF + FG and AB + BC = AC, then EF + FG = AC.

PROOF Complete each proof.

5. Given: −−− SU " −−

LR −−−

TU " −−− LN

Prove: −− ST " −−−

NR Proof:

6. Given: −− AB "

−−− CD

Prove: −−− CD "

−− AB

Proof:

US T

RL N

Statements Reasonsa. −−−

SU " −− LR , −−−

TU " −−− LN

b. c. SU = ST + TU

LR = LN + NRd. ST + TU = LN + NRe. ST + LN = LN + NR

f. ST + LN - LN = LN+ NR - LNg. h. −−

ST " −−−

NR

a. b. Definition of " segmentsc.

d. e. f.

g. Substitution Propertyh.

Statements Reasons

a. b. AB = CDc. CD = AB d.

a. Givenb. c. d. Definition of " segments

Skills PracticeProving Segment Relationships

2-7

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Chapter 2 46 Glencoe Geometry

CA B

FD

E

Complete the following proof.

1. Given: −− AB "

−−− DE

B is the midpoint of −−

AC . E is the midpoint of −−−

DF . Prove: −−−

BC " −−

EF Proof:

2. TRAVEL Refer to the figure. DeAnne knows that the distance from Grayson to Apex is the same as the distance from Redding to Pine Bluff. Prove that the distance from Grayson to Redding is equal to the distance from Apex to Pine Bluff.

Statements Reasons

a.

b. AB = DEc.

d. BC = DEe. BC = EFf.

a. Given

b. c. Definition of Midpoint

d. e. f.

Grayson Apex Redding Pine Bluff

G A R P

2-7 PracticeProving Segment Relationships

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