1. 2.7 Proving Statements about Segments

2. Definitions
Theorem:
A true statement that follows as a result of other true statements.
Two-column proof:
Most commonly used. Has numbered statements and reasons that show the logical order of an argument.

3. Definitions/Properties/ Postulates/Theorems/Formulas
Segment congruence is reflexive, symmetric, and transitive.
Examples:
Reflexive: For any segment AB, AB ≅ AB
Symmetric: If AB ≅ CD, then CD ≅ AB
Transitive: If AB ≅ CD, and CD ≅ EF, then AB ≅ EF

4. Example 1: Using Congruence
Use the diagram and the given information to complete the missing steps and reasons in the proof.
GIVEN: LK = 5, JK = 5, JK ≅ JL
PROVE: LK ≅ JL

5. ________________ LK = JK LK ≅ JK JK ≅ JL Given _________________
Statements: Reasons:
________________
LK = JK
LK ≅ JK
JK ≅ JL
Given
_________________
Transitive Property

6. Substitution Property Def. Congruent seg. Transitive Property
Statements: Reasons:
LK = 5
JK = 5
LK = JK
LK ≅ JK
JK ≅ JL
LK ≅ JL
Given
Substitution Property
Def. Congruent seg.
Transitive Property

7. Example 3: Using Segment Relationships
In the diagram, Q is the midpoint of PR. Show that PQ and QR are equal to ½ PR.
GIVEN: Q is the midpoint of PR.
PROVE: PQ = ½ PR and QR = ½ PR.

8. Definition of a midpoint Substitution Property Combine Like Terms
Statements: Reasons:
Q is the midpoint of PR.
PQ = QR
PQ + QR = PR
PQ + PQ = PR
2 ∙ PQ = PR
PQ = ½ PR
QR = ½ PR
Given
Definition of a midpoint
Segment Addition Postulate
Substitution Property
Combine Like Terms
Division property
Substitution