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. Assignment:
pp #1-16 all

4. 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.

5. NOTE: Put in the Definitions/Properties/ Postulates/Theorems/Formulas 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

6. 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

7. 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.

8. 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

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

10. 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.

11. 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.

12. 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

13. Pg. 104 – Activity—Copy a segment
Use the following steps to construct a segment that is congruent to AB.
Use a straightedge to draw a segment longer than AB. Label the point C on the new segment.
Set your compass at the length of AB.
Place the compass point at C and mark a second point D on the new segment. CD is congruent to AB.

14. Plan for next week Monday – 2.6 Notes/ HW due following class meeting.
Tues/Wed – Review Chapter 2 – Due at the end of class period
Thur/Fri – Chapter 2 Exam; Chapter 3 definitions pg Copy the following into your definitions/postulates/theorems portion of your binder: pg. 130, 137, 138, 143, 150, 157, 166, 172 – Look for the green boxes.