2 7 Proving Segment Relationships What youll learn

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 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 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