1. 2.5 Proving Statements about Segments
Mrs. Spitz
Geometry
Fall 2004

2. Standards/Objectives:
Standard 3: Students will learn and apply geometric concepts.
Objectives:
Justify statements about congruent segments.
Write reasons for steps in a proof.

3. 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.

4. Theorem and Examples Theorem 2.1 Examples:
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

5. 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

6. 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
_________________

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.

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 Distributive property
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
Distributive property
Division property
Substitution

11. Plan for next week
Thurs/Fri– Continue 2.6 Notes/ HW due following class meeting.
Mon/Tues – Review Chapter
Thur/Fri – Chapter 2 Exam; Chapter 3 definitions