1. 2.7 Proving Segment Relationships

2. Ruler Postulate Postulate 2.8 (Ruler Postulate)
The points on any line or line segment can be paired with real numbers so that, given any two points A and B on a line, A corresponds to zero, and B corresponds to a positive real number.
 (all segments have a measure.)

3. Segment Addition 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.
AB BC
A B C AC

4. Example 1: Given: PR = QS Prove the following. Prove: PQ = RS Proof:
Statements Reasons
1. Given
PR = QS 1.
2. Subtraction Property
PR – QR = QS – QR 2.
3. Segment Addition Postulate
PR – QR = PQ; QS – QR = RS 3.
4. Substitution
PQ = RS 4.

5. Your Turn:
Prove the following.
Prove:
Given:

6. Your Turn: Statements Reasons Given Transitive Property
AC = AB, AB = BX
AC = BX
CY = XD
AC + CY = BX + XD
AC + CY = AY; BX + XD = BD
AY = BD
Given
Transitive Property
Addition Property
Segment Addition Property
Substitution

7. Segment Congruence Theorem 2.2 (Segment Congruence)
Congruence of segments is reflexive, symmetric, and transitive.
Reflexive Property: AB  AB
Symmetric Property: If AB  CD, then CD  AB.
Transitive Property: If AB  CD and CD  EF, then AB  EF.

8. Example 2:
Prove the following.
Prove:
Given:

9. Example 2: Proof: Statements Reasons 1. 1. Given 2.
2. Definition of congruent segments 2.
3. Given 3.
4. Transitive Property 4.
5. Transitive Property 5.

10. Your Turn:
Prove the following.
Prove:
Given:

11. Your Turn: Statements Reasons 1. Given 2. Transitive Property 3. 4. 5.
Symmetric Property