2.7 Proving Segment
Relationships
What you’ll learn:
1. To write proofs involving segment
addition.
2. To write proofs involving segment
congruence.
Theorems
Theorem – a statement or conjecture that can
be proven true by undefined terms,
definitions, and postulates.
Theorem 2.8 – If M is the midpoint of AB, then
AMMB.
Postulate 2.8 Ruler Postulate
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.
A
B
C
Segment Congruence
Congruence of segments is reflexive, symmetric,
and transitive.
Reflexive - ABAB
Symmetric – If ABCD, then CDAB.
Transitive – If ABCD and CDEF, then ABEF.
Other properties of equality may also be used in
proofs involving segments.
Segment congruence verses equal segments.
AB=CD can be changed to ABCD by the
definition of congruent segments. (If they’re
congruent, they’re equal and vice-versa.)
Name that property
1. If PQ+ST=KL+ST, then PQ=KL
subtraction
2. If ST=UV and UV=WX, then ST=WX.
transitive
3. If LM=20 and PQ=20, then LM=PQ.
substitution
4. If D, E, and F are on the same line with E in
between D and F, then DE+EF=DF.
segment addition position
Write a 2-column proof
Given: BC=DE
Prove: AB+DE=AC
Statements
BC=DE
AB+BC=AC
AB+DE=AC
D
A
E
B
C
Reasons
given
Seg. Add. Post.
substitution
Write a 2-column proof
Given: C is the midpoint of BD, D is the midpoint of CE.
Prove: BDCE
B
Statements
1. C is the midpoint of BD, D
is the midpoint of CE.
2. BC=CD, CD=DE
3. BC=DE
4. BC+CD=BD, CD+DE=CE
5. DE+CD=BD
6. BD=CE
7. BDCE
C
D
Reasons
Given
Defn. midpoint
Transitive
Seg. Add. post.
Substitution
substitution
Defn. congruent segments
E
Homework
p. 104
12-23 all
32-44 even