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NAME ______________________________________________ DATE
2-7
____________ 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.
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.
Segment Addition
Postulate
B is between A and C if and only if AB 1 BC 5 AC.
Example
Write a two-column proof.
Given: Q is the midpoint of P
wR
w.
R is the midpoint of w
QS
w.
Prove: PR 5 QS
P
Q
R
Lesson 2-7
Ruler Postulate
S
Statements
Reasons
1. Q is the midpoint of w
PR
w.
2. PQ 5 QR
3. R is the midpoint of w
QS
w.
4. QR 5 RS
5. PQ 1 QR 5 QR 1 RS
6. PQ 1 QR 5 PR, QR 1 RS 5 QS
7. PR 5 QS
1. Given
2. Definition of midpoint
3. Given
4. Definition of midpoint
5. Addition Property
6. Segment Addition Postulate
7. Substitution
Exercises
Complete each proof.
1. Given: BC 5 DE
Prove: AB 1 DE 5 AC
©
A
B
D
E
Statements
a. BC 5 DE
Reasons
a.
b.
b. Seg. Add. Post.
c. AB 1 DE 5 AC
c.
Glencoe/McGraw-Hill
2. Given: Q is between
P and R, R is between
Q and S, PR 5 QS.
Prove: PQ 5 RS
C
Statements
R
P
S
Q
Reasons
a. Q is between
a. Given
P and R.
b.PQ 1 QR 5 PR
b.
c. R is between
c.
Q and S.
d.
d. Seg. Add. Post.
e. PR 5 QS
e.
f. PQ 1 QR 5
f.
QR 1 RS
g. PQ 1 QR 2 QR 5 g.
QR 1 RS 2 QR
h.
h. Substitution
93
Glencoe Geometry
NAME ______________________________________________ DATE
2-7
____________ PERIOD _____
Study Guide and Intervention
(continued)
Proving Segment Relationships
Segment Congruence Three properties of algebra—the Reflexive, Symmetric, and
Transitive Properties of Equality—have counterparts as properties of geometry. These
properties can be proved as a theorem. As with other theorems, the properties can then be
used to prove relationships among segments.
Segment Congruence Theorem
Congruence of segments is reflexive, symmetric, and transitive.
Reflexive Property
wB
A
w>A
wB
w
Symmetric Property
If A
wB
w>C
wD
w, then C
wD
w>A
wB
w.
Transitive Property
If A
wB
w>C
wD
w and C
wD
w>E
wF
w, then A
wB
w>E
wF
w.
Example
Write a two-column proof.
Given: w
AB
w>w
DE
w; w
BC
w>E
wF
w
Prove: w
Aw
C>w
DF
w
B
A
C
E
F
D
Statements
Reasons
1. A
wB
w>D
wE
w
2. AB 5 DE
3. w
BC
w>E
wF
w
4. BC 5 EF
5. AB 1 BC 5 DE 1 EF
6. AB 1 BC 5 AC, DE 1 EF 5 DF
7. AC 5 DF
8. w
AC
w>D
wF
w
1. Given
2. Definition of congruence of segments
3. Given
4. Definition of congruence of segments
5. Addition Property
6. Segment Addition Postulate
7. Substitution
8. Definition of congruence of segments
Exercises
Justify each statement with a property of congruence.
1. If D
wE
w>w
GH
w, then G
wH
w>D
wE
w.
2. If A
wB
w>R
wS
w and w
RS
w>W
wY
w, then A
wB
w>w
WY
w.
3. w
RS
w>R
wS
w
4. Complete the proof.
Given: P
wR
w>w
QS
w
Prove: P
wQ
w>w
RS
w
©
.
R
P
S
Q
Statements
Reasons
a. P
wR
w>w
QS
w
b. PR 5 QS
c. PQ 1 QR 5 PR
d.
e. PQ 1 QR 5 QR 1 RS
f.
g.
a.
b.
c.
d. Segment Addition Postulate
e.
f. Subtraction Property
g. Definition of congruence of segments
Glencoe/McGraw-Hill
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Glencoe Geometry
NAME ______________________________________________ DATE
2-7
____________ PERIOD _____
Skills Practice
Proving Segment Relationships
Justify each statement with a property of equality, a property of congruence, or a
postulate.
1. QA 5 QA
Lesson 2-7
wB
w>w
BC
w and B
wC
w>w
CE
w, then w
AB
w>w
CE
w.
2. If A
3. If Q is between P and R, then PR 5 PQ 1 QR.
4. If AB 1 BC 5 EF 1 FG and AB 1 BC 5 AC, then EF 1 FG 5 AC.
Complete each proof.
5. Given: w
SU
w>L
wR
w
wU
T
w>L
wN
w
Prove: S
wT
w>w
NR
w
Proof:
S
L
T
N
U
R
Statements
Reasons
a. S
wU
w>L
wR
w, w
TU
w>L
wN
w
a.
b.
b. Definition of > segments
c. SU 5 ST 1 TU
c.
LR 5 LN 1 NR
d. ST 1 TU 5 LN 1 NR
d.
e. ST 1 LN 5 LN 1 NR
e.
f. ST 1 LN 2 LN 5 LN 1 NR 2 LN
f.
g.
g. Substitution Property
h. S
wT
w>N
wR
w
h.
6. Given: w
AB
w>w
CD
w
Prove: C
wD
w>A
wB
w
Proof:
©
Statements
Reasons
a.
a. Given
b. AB 5 CD
b.
c. CD 5 AB
c.
d.
d. Definition of > segments
Glencoe/McGraw-Hill
95
Glencoe Geometry
NAME ______________________________________________ DATE
2-7
____________ PERIOD _____
Practice
Proving Segment Relationships
Complete the following proof.
w>w
DE
w
1. Given: w
AB
B is the midpoint of w
AC
w.
E is the midpoint of w
DF
w.
Prove: B
wC
w>w
EF
w
Proof:
A
B
C
E
D
F
Statements
Reasons
a.
a. Given
b. AB 5 DE
b.
c.
c. Definition of Midpoint
d. AC 5 AB 1 BC
d.
DF 5 DE 1 EF
e. AB 1 BC 5 DE 1 EF
e.
f. AB 1 BC 5 AB 1 EF
f.
g. AB 1 BC 2 AB 5 AB 1 EF 2 AB
g. Subtraction Property
h. BC 5 EF
h.
i.
i.
2. TRAVEL Refer to the figure. DeAnne knows that the
Grayson
distance from Grayson to Apex is the same as the distance
G
from Redding to Pine Bluff. Prove that the distance from
Grayson to Redding is equal to the distance from Apex to Pine Bluff.
©
Glencoe/McGraw-Hill
96
Apex Redding
A
R
Pine Bluff
P
Glencoe Geometry
NAME ______________________________________________ DATE
2-8
____________ PERIOD _____
Study Guide and Intervention
(continued)
Proving Angle Relationships
Congruent and Right Angles
Three properties of angles can be proved as theorems.
Congruence of angles is reflexive, symmetric, and transitive.
Angles supplementary to the same angle
or to congruent angles are congruent.
Angles complementary to the same angle or to
congruent angles are congruent.
C
K
1
F
B
A
2
D
E
H
G
T
N
4
3
M
J
L
Q
If /1 and /2 are supplementary to /3,
then /1 > /2.
6
5
P
R
S
If /4 and /5 are complementary to /6,
then /4 > /5.
Example
Write a two-column proof.
Given: /ABC and /CBD are complementary.
/DBE and /CBD form a right angle.
Prove: /ABC > /DBE
Statements
1. /ABC and /CBD are complementary.
/DBE and /CBD form a right angle.
2. /DBE and /CBD are complementary.
3. /ABC > /DBE
C
A
D
B
E
Reasons
1. Given
2. Complement Theorem
3. Angles complementary to the same / are >.
Exercises
Complete each proof.
1. Given: w
AB
w⊥B
wC
w;
/1 and /3 are
complementary.
Prove: /2 > /3
©
A
1 2
B
3
2. Given: /1 and /2
form a linear pair.
m/1 1 m/3 5 180
Prove: /2 > /3
E
C
D
T
L
1 2
B 3
R
P
Statements
Reasons
Statements
Reasons
a. A
wB
w⊥w
BC
w
b.
a.
b. Definition of ⊥
a. /1 and /2 form
a linear pair.
m/1 1 m/3 5 180
a. Given
c. m/1 1 m/2 5
m/ABC
d. /1 and /2 form
a rt /.
e. /1 and /2 are
compl.
f.
c.
b.
b. Suppl.
Theorem
d.
c. /1 is suppl.
to /3.
c.
e.
d.
d. ? suppl. to the
same / are >.
g. /2 > /3
g.
Glencoe/McGraw-Hill
f. Given
100
Glencoe Geometry