Chapter 2.7
Proving Segment Relationships
Objective: Practice using proofs for geometric relationships
by starting with segments
Spi.1.4
Check.4.3
Use definitions, basic postulates, and theorems about points, lines, angles, and planes to
write/complete proofs and/or to solve problems.
Use definitions, basic postulates, and theorems about points, lines, angles, and planes to
write/complete proofs
Add to your
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Geometric Properties

Postulate 2.8 (Ruler Postulate)


Postulate 2.9 (Segment Addition 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 zeros and B corresponds to a positive real number.
If B is between A and C, then AB + BC = AC or
If AB + BC = AC, then B is between A and C
AB
A
Theorem 2.2

BC
B
AC
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
It's not what you look at that matters, it's what you see. Henry David Thoreau
C
Use paper to solve
Given BC = DE
 Prove AB + DE = AC


1.
2.
3.
Statements
BC = DE
AB + BC = AC
AB + DE = AC

1.
2.
3.
Reasons
Given
Segment Addition Postulate
Substitution
Use paper to solve
Given PR  QS
 Prove PQ  RS


1.
2.
3.
4.
5.
6.
7.
8.
Statements
PR  QS
PR = QS
PQ + QR = PR
QR + RS = QS
PQ + QR = QR + RS
PQ = RS
PQ  RS

1.
2.
3.
4.
5.
6.
7.
Reasons
Given
Definition of Congruence
Segment Addition Postulate
Segment Addition Postulate
Substitution
Subtraction
Definition of Congruence
Proof with Segment Addition
process
Prove the following:
Given: PQ = RS
Prove: PR = QS
Statements
1. PQ = RS
P
1.
Q
R
Reasons
Given
2.
PQ + QR = QR + RS
2.
Addition Property
3.
PQ + QR = PR and
3.
Segment Addition Postulate
4.
Substitution
QR + RS = QS
4.
PR = QS
S
Proof with Segment Addition
process
Prove the following:
Given: PR = QS
Prove: PQ = RS
Statements
1. PR = QS
P
1.
Q
R
Reasons
Given
2.
PR - QR = QS - QR
2.
Subtraction Property
3.
PR - QR = PQ and
3.
Segment Addition Postulate
4.
Substitution
QS - QR = RS
4.
PQ = RS
S
Proof with Segment Congruence
process
J
Prove the following:
Given: JK  KL, HJ  GH, KL  HJ
Prove: GH  JK
Statements
1. JK  KL, KL  HJ
1.
Reasons
Given
K
L
H
2.
JK  HJ
2.
Transitive Property
3.
HJ  GH
3.
Given
4.
JK  GH
4.
Transitive Property
5.
GH  JK
5.
Symmetric Property
G
Prove the following.
Given: AC = AB
AB = BX
CY = XD
Prove: AY = BD
Which reason correctly completes the proof?
Proof:
Statements
Reasons
1. AC = AB, AB = BX
1. Given
2. AC = BX
2. Transitive Property
3. CY = XD
3. Given
4. AC + CY = BX + XD
4. Addition Property
5. AC + CY = AY;
BX + XD = BD
? Addition Postulate
Segment
5. ________________
6. AY = BD
6. Substitution
Practice Assignment

Page 145, 4-16 even