2.7 proving segment relationships what you’ll learn: 1.to write proofs involving segment addition....
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2.7 Proving Segment RelationshipsWhat you’ll learn:
1. To write proofs involving segment addition.
2. To write proofs involving segment congruence.
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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
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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.)
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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
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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
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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
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Homeworkp. 104
12-23 all32-44 even