6.3 Proving Quadrilaterals are Parallelograms

1. 6.3 Proving Quadrilaterals are Parallelograms Standard: 7.0 & 17.0

2. Theorem 6.5: If both pairs of opposite sides of a quadrilateral are congruent, then the quadrilateral is a parallelogram. Theorems ABCD is a parallelogram.

3. Theorem 6.6: If both pairs of opposite angles of a quadrilateral are congruent, then the quadrilateral is a parallelogram. Theorems ABCD is a parallelogram.

4. Theorem 6.7: If the diagonals of a quadrilateral bisect each other, then the quadrilateral is a parallelogram. Theorems ABCD is a parallelogram.

5. Statements: AB ≅ CD, AD ≅ CB. AC ≅ AC ∆ABC ≅ ∆CDA BAC ≅ DCA, DAC ≅ BCA AB║CD, AD ║CB. ABCD is a  Reasons: Given Ex. 1: Proof of Theorem 6.5

6. Statements: AB ≅ CD, AD ≅ CB. AC ≅ AC ∆ABC ≅ ∆CDA BAC ≅ DCA, DAC ≅ BCA AB║CD, AD ║CB. ABCD is a  Reasons: Given Reflexive Prop. of Congruence Ex. 1: Proof of Theorem 6.5

7. Statements: AB ≅ CD, AD ≅ CB. AC ≅ AC ∆ABC ≅ ∆CDA BAC ≅ DCA, DAC ≅ BCA AB║CD, AD ║CB. ABCD is a  Reasons: Given Reflexive Prop. of Congruence SSS Congruence Postulate Ex. 1: Proof of Theorem 6.5

8. Statements: AB ≅ CD, AD ≅ CB. AC ≅ AC ∆ABC ≅ ∆CDA BAC ≅ DCA, DAC ≅ BCA AB║CD, AD ║CB. ABCD is a  Reasons: Given Reflexive Prop. of Congruence SSS Congruence Postulate CPCTC Ex. 1: Proof of Theorem 6.5

9. Statements: AB ≅ CD, AD ≅ CB. AC ≅ AC ∆ABC ≅ ∆CDA BAC ≅ DCA, DAC ≅ BCA AB║CD, AD ║CB. ABCD is a  Reasons: Given Reflexive Prop. of Congruence SSS Congruence Postulate CPCTC Alternate Interior s Converse Ex. 1: Proof of Theorem 6.5

10. Statements: AB ≅ CD, AD ≅ CB. AC ≅ AC ∆ABC ≅ ∆CDA BAC ≅ DCA, DAC ≅ BCA AB║CD, AD ║CB. ABCD is a  Reasons: Given Reflexive Prop. of Congruence SSS Congruence Postulate CPCTC Alternate Interior s Converse Def. of a parallelogram. Ex. 1: Proof of Theorem 6.5

11. Another Theorem ~ • Theorem 6.8—If one pair of opposite sides of a quadrilateral are congruent and parallel, then the quadrilateral is a parallelogram. • ABCD is a parallelogram. B C A D

12. Statements: BC ║DA DAC ≅ BCA AC ≅ AC BC ≅ DA ∆BAC ≅ ∆DCA AB ≅ CD ABCD is a  Reasons: 1. Given Ex. 3: Proof of Theorem 6.8Given: BC║DA, BC ≅ DAProve: ABCD is a 

13. Statements: BC ║DA DAC ≅ BCA AC ≅ AC BC ≅ DA ∆BAC ≅ ∆DCA AB ≅ CD ABCD is a  Reasons: Given Alt. Int. s Thm. Ex. 3: Proof of Theorem 6.8Given: BC║DA, BC ≅ DAProve: ABCD is a 

14. Statements: BC ║DA DAC ≅ BCA AC ≅ AC BC ≅ DA ∆BAC ≅ ∆DCA AB ≅ CD ABCD is a  Reasons: Given Alt. Int. s Thm. Reflexive Property Ex. 3: Proof of Theorem 6.8Given: BC║DA, BC ≅ DAProve: ABCD is a 

15. Statements: BC ║DA DAC ≅ BCA AC ≅ AC BC ≅ DA ∆BAC ≅ ∆DCA AB ≅ CD ABCD is a  Reasons: Given Alt. Int. s Thm. Reflexive Property Given Ex. 3: Proof of Theorem 6.8Given: BC║DA, BC ≅ DAProve: ABCD is a 

16. Statements: BC ║DA DAC ≅ BCA AC ≅ AC BC ≅ DA ∆BAC ≅ ∆DCA AB ≅ CD ABCD is a  Reasons: Given Alt. Int. s Thm. Reflexive Property Given SAS Congruence Post. Ex. 3: Proof of Theorem 6.8Given: BC║DA, BC ≅ DAProve: ABCD is a 

17. Statements: BC ║DA DAC ≅ BCA AC ≅ AC BC ≅ DA ∆BAC ≅ ∆DCA AB ≅ CD ABCD is a  Reasons: Given Alt. Int. s Thm. Reflexive Property Given SAS Congruence Post. CPCTC Ex. 3: Proof of Theorem 6.8Given: BC║DA, BC ≅ DAProve: ABCD is a 

18. Statements: BC ║DA DAC ≅ BCA AC ≅ AC BC ≅ DA ∆BAC ≅ ∆DCA AB ≅ CD ABCD is a  Reasons: Given Alt. Int. s Thm. Reflexive Property Given SAS Congruence Post. CPCTC If opp. sides of a quad. are ≅, then it is a. Ex. 3: Proof of Theorem 6.8Given: BC║DA, BC ≅ DAProve: ABCD is a 

19. Assignment • 324-5 # 1-9, 12, 14-18