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Given: ZY || WX; WX  ZY Prove:  W   Y

WARM UP - Complete the Proof. Z. Y. Given: ZY || WX; WX  ZY Prove:  W   Y. W. X. 1. WX  ZY, ZY || WX. 1. Given. 2. If lines are parallel, alternate interior angles are congruent. 2. YZX  WXZ. 3. ZX  ZX. 3. Reflexive Property. 4. Δ YZX  Δ WXZ.

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Given: ZY || WX; WX  ZY Prove:  W   Y

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  1. WARM UP- Complete the Proof. Z Y Given: ZY || WX; WX  ZY Prove: W Y W X 1. WX  ZY, ZY || WX 1. Given 2. If lines are parallel, alternate interior angles are congruent. 2. YZX  WXZ 3. ZX  ZX 3. Reflexive Property 4. ΔYZX ΔWXZ 4. SAS Postulate 5. W  Y 5. CPCTC

  2. Section 5-1 Properties of Parallelograms

  3. PARALLELOGRAM Definition – A quadrilateral with both pairs of opposite sides parallel.

  4. Theorem 5-1: Opposite sides of a parallelogram are congruent. E F H G EF  GH, FG  EH

  5. F PROOF OF THEOREM 5-1: E 1 2 Given: EFGH Prove: EF  GH, FG  EH 3 4 G H 1. EFGH 1. Given 2. EF || GH, FG || HE 2. Def. of parallelogram 3. If lines are parallel, then alternate interior angles are congruent. 3. 1 4, 2 3 4. FH FH 4. Reflexive Property 5. ΔEFH ΔGHF 5. ASA Postulate 6. EF GH; FG  HE 6. CPCTC

  6. Theorem 5-2: Opposite angles of a parallelogram are congruent. E F H G F  H, E  G

  7. F E PROOF OF THEOREM 5-2: 4 1 Given: EFGH Prove: E  G 2 3 G H 1. EFGH 1. Given 2. EF || HG, EH || FG 2. Def. of parallelogram 3. If lines are parallel, then alternate interior angles are congruent. 3. 4 3, 2 1 4. FH FH 4. Reflexive Property 5. ΔEFH ΔGHF 5. ASA Postulate 6. E  G 6. CPCTC

  8. How can we prove the other opposite angles are congruent using information from the previous proof? F E Given: EFGH, E  G Prove: H  F G H 1. E  G, EFGH 1. Given 2. EF || HG 2. Def. of parallelogram 3. Eand H are supplementary 3. If lines are parallel, same-side interior angles are supplementary. Fand G are supplementary 4. mE + mH = 180 4. Definition of Supp. Angles mF + mG = 180 5. mE + mH = mF + mG 5. Substitution 6. mH = mF, H  F 6. Subtraction

  9. Consecutive angles of a parallelogram are supplementary . E F H G E and H are supplementary E and F are supplementary F and G are supplementary H and G are supplementary

  10. Theorem 5-3: Diagonals of a parallelogram bisect each other. Q R M T S QM  MS, RM  MT

  11. Q PROOF OF THEOREM 5-3: R 1 3 Given: QRST Prove: QS and RT bisect each other M 4 2 S T 1. QRST 1. Given 2. QR || ST 2. Def. of parallelogram 3. If lines are parallel, then alternate interior angles are congruent. 3. 1 2, 3 4 4. QR ST 4. Opposite sides of a parallelogram are congruent. 5. ΔQRM ΔSTM 5. ASA Postulate 6. QM SM; RM  TM 6. CPCTC 7. M is the midpoint of QS and RT 7. Definition of a Midpoint 8. QS and RT bisect each other 8. Def. of Segment Bisector

  12. Review of Properties Name all the properties of a parallelogram. • 2 pairs of opposite sides are parallel • 2 pairs of opposite sides are congruent • 2 pairs of opposite angles are congruent • Diagonals bisect each other • Consecutive angles are supplementary

  13. CLASS WORK Complete 5-1 Class Work worksheet and turn it in when you are finished.

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