NAME

2-7

DATE

PERIOD

Study Guide and Intervention Proving Segment Relationships

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. Ruler Postulate

The 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.

Example

Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.

Write a two-column proof. −− Given: Q is the midpoint of PR. −−− R is the midpoint of QS. Prove: PR = QS Proof:

P

Q

R

S

Statements −− 1. Q is the midpoint of PR. 2. PQ = QR −−− 3. R is the midpoint of QS. 4. QR = RS 5. PQ + QR = QR + RS

Reasons 1. Given 2. Definition of midpoint 3. Given 4. Definition of midpoint 5. Addition Property

6. PQ = RS

6. Transitive Property

7. PQ + QR = PR, QR + RS = QS 8. PR = QS

7. Segment Addition Postulate 8. Substitution

Exercises 2. Given: Q is between P and R, R is between R S Q Q and S, PR = QS. P Prove: PQ = RS Proof: Statements Reasons 1. 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 + RS 7. PQ + QR - QR = 7. QR + RS - QR 8. 8. Substitution

Complete each proof. 1. Given: BC = DE Prove: AB + DE = AC A B Proof: Statements 1. BC = DE 2. 3. AB + DE = AC

Chapter 2

C D

E

Reasons 1. 2. Seg. Add. Post. 3.

27

Glencoe Geometry

NAME

2-7

DATE

PERIOD

Study Guide and Intervention

(continued)

Proving Segment Relationships 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. Reflexive Property Symmetric Property Transitive Property

−− −− AB  AB −−− −− −− −−− If AB  CD, then CD  AB. −−− −− −− −−− −− −− If AB  CD and CD  EF, then AB  EF.

Write a two-column Example −− −−− −−− −− Given: AB  DE; BC  EF −− −−− Prove: AC  DF Proof: Statements −− −−− 1. AB  DE 2. AB = DE −−− −− 3. BC  EF 4. BC = EF 5. AB + BC = DE + EF

proof. B

C

A

E

F

D

Reasons 1. Given 2. Definition of congruence of segments 3. Given 4. Definition of congruence of segments 5. Addition Property 6. Segment Addition Postulate

7. AC = DF −− −−− 8. AC  DF

7. Substitution 8. Definition of congruence of segments

Exercises Justify 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: Statements −− −−− a. PR  QS b. PR = QS c. PQ + QR = PR d. e. PQ + QR = QR + RS f. g. Chapter 2

R P

S

Q

Reasons a. b. c. d. Segment Addition Postulate e. f. Subtraction Property g. Definition of congruence of segments

28

Glencoe Geometry

Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.

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