1. Sect. 2.5 Proving Statements about Segments.
Goal Properties of Congruent Segments
Goal Using Congruence of Segments

2. Properties of Congruent Segments
Definition:
Theorem: A true statement that follows as a result of other true statements.
proved through deductive reasoning using definitions, postulates, and undefined terms

3. Definition: Definition of Congruent Segments AB = CD if and only if
Properties of Congruent Segments
Definition:
Definition of Congruent Segments
AB = CD if and only if
Definition of Congruent Angles
mA = m B if and only if A  B

4. Properties of Segment Congruence
Properties of Congruent Segments
Theorem 2.1
Properties of Segment Congruence
Reflexive: For any Segment
Symmetric: If , then
Transitive: If and , then

5. Definition: Proofs are often organized in a TWO COLUMN FORMAT.
Properties of Congruent Segments
Definition:
Proofs are often organized in a TWO COLUMN FORMAT. 
The left column contains the statements or steps of the proof.
The right column contains how you know the corresponding statement to be true.  
Reasons for knowing something would include postulates, rules, givens, properties, or theorems. 
Typically the structure of a proof begins by working from what you are given to systematically proving the desired outcome.

6. Properties of Congruent Segments
Definition:
Paragraph Proof: A two-column proof written in paragraph form.

7. Five essential steps to construct any proof.
Properties of Congruent Segments
Five essential steps to construct any proof.
1. State the theorem to be proved.
2. List the Given Information.
If possible, draw a diagram to
illustrate the given information.
4. State what is to be proved.
Develop a system of deductive
reasoning.

8. Example 1 Given: EF = GH Prove: Statements Reasons 1. EF = GH 1. Given
Using Congruence of Segments
Example 1
Given: EF = GH
Prove:
Statements
Reasons
1. EF = GH
1. Given
2. EF + FG = GH + FG
2. Addition Prop of Equality
EG = EF + FG
FH = GH + FG
3. Segment Addition Postulate
4. EG = FH
4. Substitution
5. Definition of Congruent Segments. 5.

9. Statements Reasons 1. ST = WX ; 1. 2. RT = WY 2. 3. RT = RS + ST
Using Congruence of Segments
Example 2
Given: ST = WX;
Prove:
Statements
Reasons
1. ST = WX ; 1.
2. RT = WY 2.
3. RT = RS + ST
WY = WX + XY 3.
4. RS + ST = WX + XY 4.
5. RS + ST = ST + XY 5.
6. RS = XY 6. 7.
Given
Def  Segs
SAP
Substitution
Subtraction Property
Def  Segments

10. Given: RX = MX; X is midpoint of Prove: XN = RX
Using Congruence of Segments
Given: RX = MX; X is midpoint of
Prove: XN = RX
Statements
Reasons
1.RX = MX; X is midpoint of MN
RX = MX
2. MX = XN
3. RX = XN
Given
Def MdPt
Subst or Transitive

11. Homework
, 16-18, 21-26