1. 2.5 Proving Statements about Segments

2. Goal 1: Properties of Congruent Segments
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. NOTE: Put in the Postulates/Theorems/Properties portion of your notebook
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: Symmetric Property of Segment Congruence
Given: PQ ≅ XY
Prove XY ≅ PQ
Statements:
PQ ≅ XY
PQ = XY
XY = PQ
XY ≅ PQ
Reasons:
Given
Definition of congruent segments
Symmetric Property of Equality

5. Paragraph Proof
A proof can be written in paragraph form. It is as follows:
You are given that PQ ≅ to XY. By the definition of congruent segments, PQ = XY. By the symmetric property of equality, XY = PQ. Therefore, by the definition of congruent segments, it follows that XY ≅ PQ.

6. Goal 2: Using Congruence of Segments
Example 2 – 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

7. ________________ LK = JK LK ≅ JK JK ≅ JL Given Transitive Property
Statements: Reasons:
________________
LK = JK
LK ≅ JK
JK ≅ JL
Given
Transitive Property
_________________
GIVEN: LK = 5, JK = 5, JK ≅ JL
PROVE: LK ≅ JL

8. LK = 5 JK = 5 LK = JK LK ≅ JK JK ≅ JL LK ≅ JL Given
Statements: Reasons:
LK = 5
JK = 5
LK = JK
LK ≅ JK
JK ≅ JL
LK ≅ JL
Given
Transitive Property
Def. Congruent seg.
GIVEN: LK = 5, JK = 5, JK ≅ JL
PROVE: LK ≅ JL

9. 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.

10. Definition of a midpoint Substitution Property Simplify
Statements: Reasons:
Q is the midpoint of PR.
PQ = QR
PQ + QR = PR
PQ + PQ = PR
2PQ = PR
PQ = ½ PR
QR = ½ PR
Given
Definition of a midpoint
Segment Addition Postulate
Substitution Property
Simplify
Division property
Substitution
GIVEN: Q is the midpoint of PR.
PROVE: PQ = ½ PR and QR = ½ PR.