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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 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. Given: EF = GH Prove EG ≌ FH #1. EF = GH E F G #1. Given H Given: EF = GH Prove EG ≌ FH #1. EF = GH #2. FG = FG E F G H #1. Given #2. Reflexive Prop. Given: EF = GH Prove EG ≌ FH E #1. EF = GH #2. FG = FG #3. EF + FG = GH + FG F G H #1. Given #2. Reflexive Prop. #3. Add. Prop. Given: EF = GH Prove EG ≌ FH #1. #2. #3. #4. E EF = GH FG = FG EF + FG = GH + FG EG = EF + FG FH = FG + GH F G H #1. Given #2. Reflexive Prop. #3. Add. Prop. #4. Given: EF = GH Prove EG ≌ FH #1. #2. #3. #4. E EF = GH FG = FG EF + FG = GH + FG EG = EF + FG FH = FG + GH F #1. #2. #3. #4. G H Given Reflexive Prop. Add. Prop. Segment Add. Given: EF = GH Prove EG ≌ FH #1. #2. #3. #4. E EF = GH FG = FG EF + FG = GH + FG EG = EF + FG FH = FG + GH #5. EG = FH F #1. #2. #3. #4. G H Given Reflexive Prop. Add. Prop. Segment Add. #5. Subst. Prop. Given: EF = GH Prove EG ≌ FH #1. #2. #3. #4. E EF = GH FG = FG EF + FG = GH + FG EG = EF + FG FH = FG + GH #5. EG = FH #6. EG ≌ FH F #1. #2. #3. #4. G H Given Reflexive Prop. Add. Prop. Segment Add. #5. Subst. Prop. #6. Def. of ≌ Given: RT ≌ WY; ST = WX Prove: RS ≌ XY W #1. RT ≌ WY R X #1. Given S Y T Given: RT ≌ WY; ST = WX Prove: RS ≌ XY #1. RT ≌ WY #2. RT = WY R W #1. Given #2. Def. of ≌ S X T Y Given: RT ≌ WY; ST = WX Prove: RS ≌ XY #1. RT ≌ WY #2. RT = WY #3. RT = RS + ST WY = WX + XY R W S X #1. Given #2. Def. of ≌ #3. Segment Add. T Y Given: RT ≌ WY; ST = WX Prove: RS ≌ XY #1. RT ≌ WY #1. #2. RT = WY #2. #3. RT = RS + ST #3. WY = WX + XY #4. RS + ST = WX + XY R W S X T Y Given Def. of ≌ Segment Add. #4. Subst. Prop. Given: RT ≌ WY; ST = WX Prove: RS ≌ XY #1. RT ≌ WY #1. #2. RT = WY #2. #3. RT = RS + ST #3. WY = WX + XY #4. RS + ST = WX + XY #5. ST = WX #5. R W S X T Y Given Def. of ≌ Segment Add. #4. Subst. Prop. Given Given: RT ≌ WY; ST = WX Prove: RS ≌ XY #1. RT ≌ WY #1. #2. RT = WY #2. #3. RT = RS + ST #3. WY = WX + XY #4. RS + ST = WX + XY #5. ST = WX #5. #6. RS = XY #6. R W S X T Y Given Def. of ≌ Segment Add. #4. Subst. Prop. Given Subtract. Prop. Given: RT ≌ WY; ST = WX Prove: RS ≌ XY #1. RT ≌ WY #1. #2. RT = WY #2. #3. RT = RS + ST #3. WY = WX + XY #4. RS + ST = WX + XY #5. ST = WX #5. #6. RS = XY #6. #7. RS ≌ XY #7. R W S X T Y Given Def. of ≌ Segment Add. #4. Subst. Prop. Given Subtract. Prop. Def. of ≌ Given: Prove: x is the midpoint of MN and MX = RX XN = RX #1. x is the midpoint of MN #1. Given Given: Prove: x is the midpoint of MN and MX = RX XN = RX #1. x is the midpoint of MN #1. Given #2. XN = MX #2. Def. of midpoint Given: Prove: x is the midpoint of MN and MX = RX XN = RX #1. x is the midpoint of MN #1. Given #2. XN = MX #2. Def. of midpoint #3. MX = RX #3. Given Given: Prove: #1. #2. #3. #4. x is the midpoint of MN and MX = RX XN = RX x is the midpoint of MN XN = MX #2. MX = RX #3. XN = RX #4. #1. Given Def. of midpoint Given Transitive Prop. Something with Numbers If AB = BC and BC = CD, then find BC A D 3X – 1 2X + 3 B C Page 105 # 6 - 11 Page 106 # 16 – 18, 21 - 22