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 BC

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 StatementsReasons BC=DEgiven AB+BC=ACSeg. Add. Post. AB+DE=ACsubstitution A B C D E

Write a 2-column proof Given: C is the midpoint of BD, D is the midpoint of CE. Prove: BD  CE 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 Reasons Given Defn. midpoint Transitive Seg. Add. post. Substitution substitution Defn. congruent segments BCDE

Homework p all even