2.7 Proving Segment Relationships

1. 2.7 Proving Segment Relationships What you’ll learn: To write proofs involving segment addition. To write proofs involving segment congruence.

2. 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

3. 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.)

4. Name that property • If PQ+ST=KL+ST, then PQ=KL subtraction • If ST=UV and UV=WX, then ST=WX. transitive • If LM=20 and PQ=20, then LM=PQ. substitution • If D, E, and F are on the same line with E in between D and F, then DE+EF=DF. segment addition position

5. Write a 2-column proof E Given: BC=DE Prove: AB+DE=AC Statements Reasons BC=DE given AB+BC=AC Seg. Add. Post. AB+DE=AC substitution D C B A

6. Write a 2-column proofGiven: C is the midpoint of BD, D is the midpoint of CE.Prove: BDCE B C D E Statements Reasons Given Defn. midpoint Transitive Seg. Add. post. Substitution substitution Defn. congruent segments • C is the midpoint of BD, D is the midpoint of CE. • BC=CD, CD=DE • BC=DE • BC+CD=BD, CD+DE=CE • DE+CD=BD • BD=CE • BDCE

7. Homeworkp. 10412-23 all32-44 even