2.7 Proving Segment RelationshipsWhat you’ll learn:

1. To write proofs involving segment addition.

2. To write proofs involving segment congruence.

TheoremsTheorem – 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 AMMB.

Postulate 2.8 Ruler PostulatePostulate 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 - ABABSymmetric – If ABCD, then CDAB.Transitive – If ABCD and CDEF, then ABEF.Other properties of equality may also be used in

proofs involving segments.Segment congruence verses equal segments.

AB=CD can be changed to ABCD by the definition of congruent segments. (If they’re congruent, they’re equal and vice-versa.)

Name that property1. If PQ+ST=KL+ST, then PQ=KL

subtraction2. If ST=UV and UV=WX, then ST=WX.

transitive3. If LM=20 and PQ=20, then LM=PQ.

substitution4. 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=DEProve: AB+DE=ACStatements ReasonsBC=DE givenAB+BC=AC Seg. Add.

Post.AB+DE=AC substitution

AB

C

DE

Write a 2-column proofGiven: C is the midpoint of BD, D is the midpoint of CE.

Prove: BDCE

Statements1. C is the midpoint of BD, D

is the midpoint of CE.2. BC=CD, CD=DE3. BC=DE4. BC+CD=BD, CD+DE=CE5. DE+CD=BD6. BD=CE7. BDCE

ReasonsGiven

Defn. midpointTransitiveSeg. Add. post.SubstitutionsubstitutionDefn. congruent segments

B C D E

Homeworkp. 104

12-23 all32-44 even