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Understanding Line Segment Congruence Theorems and Proof Writing Techniques

This resource covers fundamental concepts of line segment congruence, including three essential properties: reflexive, symmetric, and transitive. It provides a structured approach to writing proofs, particularly through two-column methods. Includes practical examples to illustrate proving segment congruences based on given information, with an emphasis on adding and substituting segment lengths. Ideal for students and educators seeking to deepen their understanding of geometric proofs and theorems related to segment congruence.

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Understanding Line Segment Congruence Theorems and Proof Writing Techniques

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  1. 2.5 Proving Statements about Line Segments

  2. Theorems are statements that can be proved Theorem 2.1 Properties of Segment Congruence Reflexive AB ≌ AB All shapes are ≌ to them self Symmetric If AB ≌ CD, then CD ≌ AB Transitive If AB ≌ CD and CD ≌ EF, then AB ≌ EF

  3. How to write a Proof Proofs are formal statements with a conclusion based on given information. One type of proof is a two column proof. One column with statements numbered; the other column reasons that are numbered.

  4. Given: EF = GH Prove EG ≌ FH E F G H #1. EF = GH #1. Given

  5. Given: EF = GH Prove EG ≌ FH E F G H #1. EF = GH #1. Given #2. FG = FG #2. Reflexive Prop.

  6. Given: EF = GH Prove EG ≌ FH E F G H #1. EF = GH #1. Given #2. FG = FG #2. Reflexive Prop. #3. EF + FG = GH + FG #3. Add. Prop.

  7. Given: EF = GH Prove EG ≌ FH E F G H #1. EF = GH #1. Given #2. FG = FG #2. Reflexive Prop. #3. EF + FG = GH + FG #3. Add. Prop. #4. EG = EF + FG #4. FH = FG + GH

  8. Given: EF = GH Prove EG ≌ FH E F G H #1. EF = GH #1. Given #2. FG = FG #2. Reflexive Prop. #3. EF + FG = GH + FG #3. Add. Prop. #4. EG = EF + FG #4. Segment Add. FH = FG + GH

  9. Given: EF = GH Prove EG ≌ FH E F G H #1. EF = GH #1. Given #2. FG = FG #2. Reflexive Prop. #3. EF + FG = GH + FG #3. Add. Prop. #4. EG = EF + FG #4. Segment Add. FH = FG + GH #5. EG = FH #5. Subst. Prop.

  10. Given: EF = GH Prove EG ≌ FH E F G H #1. EF = GH #1. Given #2. FG = FG #2. Reflexive Prop. #3. EF + FG = GH + FG #3. Add. Prop. #4. EG = EF + FG #4. Segment Add. FH = FG + GH #5. EG = FH #5. Subst. Prop. #6. EG ≌ FH #6. Def. of ≌

  11. Given: RT ≌ WY; ST = WXR S TProve: RS ≌ XYW X Y #1. RT ≌ WY #1. Given

  12. Given: RT ≌ WY; ST = WX R S TProve: RS ≌ XY W X Y #1. RT ≌ WY #1. Given #2. RT = WY #2. Def. of ≌

  13. Given: RT ≌ WY; ST = WX R S TProve: RS ≌ XY W X Y #1. RT ≌ WY #1. Given #2. RT = WY #2. Def. of ≌ #3. RT = RS + ST #3. Segment Add. WY = WX + XY

  14. Given: RT ≌ WY; ST = WX R S TProve: RS ≌ XY W X Y #1. RT ≌ WY #1. Given #2. RT = WY #2. Def. of ≌ #3. RT = RS + ST #3. Segment Add. WY = WX + XY #4. RS + ST = WX + XY #4. Subst. Prop.

  15. Given: RT ≌ WY; ST = WX R S TProve: RS ≌ XY W X Y #1. RT ≌ WY #1. Given #2. RT = WY #2. Def. of ≌ #3. RT = RS + ST #3. Segment Add. WY = WX + XY #4. RS + ST = WX + XY #4. Subst. Prop. #5. ST = WX #5. Given

  16. Given: RT ≌ WY; ST = WX R S TProve: RS ≌ XY W X Y #1. RT ≌ WY #1. Given #2. RT = WY #2. Def. of ≌ #3. RT = RS + ST #3. Segment Add. WY = WX + XY #4. RS + ST = WX + XY #4. Subst. Prop. #5. ST = WX #5. Given #6. RS = XY #6. Subtract. Prop.

  17. Given: RT ≌ WY; ST = WX R S TProve: RS ≌ XY W X Y #1. RT ≌ WY #1. Given #2. RT = WY #2. Def. of ≌ #3. RT = RS + ST #3. Segment Add. WY = WX + XY #4. RS + ST = WX + XY #4. Subst. Prop. #5. ST = WX #5. Given #6. RS = XY #6. Subtract. Prop. #7. RS ≌ XY #7. Def. of ≌

  18. Given: x is the midpoint of MN and MX = RXProve: XN = RX #1. x is the midpoint of MN #1. Given

  19. Given: x is the midpoint of MN and MX = RXProve: XN = RX #1. x is the midpoint of MN #1. Given #2. XN = MX #2. Def. of midpoint

  20. Given: x is the midpoint of MN and MX = RXProve: XN = RX #1. x is the midpoint of MN #1. Given #2. XN = MX #2. Def. of midpoint #3. MX = RX #3. Given

  21. Given: x is the midpoint of MN and MX = RXProve: XN = RX #1. x is the midpoint of MN #1. Given #2. XN = MX #2. Def. of midpoint #3. MX = RX #3. Given #4. XN = RX #4. Transitive Prop.

  22. Something with Numbers If AB = BC and BC = CD, then find BC A D 3X – 1 2X + 3 B C

  23. Homework Page 105 # 6 - 11

  24. Homework Page 106 # 16 – 18, 21 - 22

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