Sect. 2.5 Proving Statements about Segments. Goal 1 Properties of Congruent Segments Goal 2 Using Congruence of Segments

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

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

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

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.

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

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.

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.

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

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

Homework 2.5 8-11, 16-18, 21-26