6.3 Proving Quadrilaterals are Parallelograms Standard: 7.0 & 17.0

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

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

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

Ex. 1: Proof of Theorem 6.5 Statements: 1. AB ≅ CD, AD ≅ CB. 2. AC ≅ AC 3. ∆ABC ≅ ∆CDA 4.  BAC ≅  DCA,  DAC ≅  BCA 5. AB ║CD, AD ║CB. 6. ABCD is a  Reasons: 1. Given

Ex. 1: Proof of Theorem 6.5 Statements: 1. AB ≅ CD, AD ≅ CB. 2. AC ≅ AC 3. ∆ABC ≅ ∆CDA 4.  BAC ≅  DCA,  DAC ≅  BCA 5. AB ║CD, AD ║CB. 6. ABCD is a  Reasons: 1. Given 2. Reflexive Prop. of Congruence

Ex. 1: Proof of Theorem 6.5 Statements: 1. AB ≅ CD, AD ≅ CB. 2. AC ≅ AC 3. ∆ABC ≅ ∆CDA 4.  BAC ≅  DCA,  DAC ≅  BCA 5. AB ║CD, AD ║CB. 6. ABCD is a  Reasons: 1. Given 2. Reflexive Prop. of Congruence 3. SSS Congruence Postulate

Ex. 1: Proof of Theorem 6.5 Statements: 1. AB ≅ CD, AD ≅ CB. 2. AC ≅ AC 3. ∆ABC ≅ ∆CDA 4.  BAC ≅  DCA,  DAC ≅  BCA 5. AB ║CD, AD ║CB. 6. ABCD is a  Reasons: 1. Given 2. Reflexive Prop. of Congruence 3. SSS Congruence Postulate 4. CPCTC

Ex. 1: Proof of Theorem 6.5 Statements: 1. AB ≅ CD, AD ≅ CB. 2. AC ≅ AC 3. ∆ABC ≅ ∆CDA 4.  BAC ≅  DCA,  DAC ≅  BCA 5. AB ║CD, AD ║CB. 6. ABCD is a  Reasons: 1. Given 2. Reflexive Prop. of Congruence 3. SSS Congruence Postulate 4. CPCTC 5. Alternate Interior  s Converse

Ex. 1: Proof of Theorem 6.5 Statements: 1. AB ≅ CD, AD ≅ CB. 2. AC ≅ AC 3. ∆ABC ≅ ∆CDA 4.  BAC ≅  DCA,  DAC ≅  BCA 5. AB ║CD, AD ║CB. 6. ABCD is a  Reasons: 1. Given 2. Reflexive Prop. of Congruence 3. SSS Congruence Postulate 4. CPCTC 5. Alternate Interior  s Converse 6. Def. of a parallelogram.

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. A BC D

Ex. 3: Proof of Theorem 6.8 Given: BC ║DA, BC ≅ DA Prove: ABCD is a  Statements: 1. BC ║DA 2.  DAC ≅  BCA  AC ≅ AC  BC ≅ DA 5. ∆BAC ≅ ∆DCA 6. AB ≅ CD  ABCD is a  Reasons: 1. Given

Ex. 3: Proof of Theorem 6.8 Given: BC ║DA, BC ≅ DA Prove: ABCD is a  Statements: 1. BC ║DA 2.  DAC ≅  BCA  AC ≅ AC  BC ≅ DA 5. ∆BAC ≅ ∆DCA 6. AB ≅ CD  ABCD is a  Reasons: 1. Given 2. Alt. Int.  s Thm.

Ex. 3: Proof of Theorem 6.8 Given: BC ║DA, BC ≅ DA Prove: ABCD is a  Statements: 1. BC ║DA 2.  DAC ≅  BCA  AC ≅ AC  BC ≅ DA 5. ∆BAC ≅ ∆DCA 6. AB ≅ CD  ABCD is a  Reasons: 1. Given 2. Alt. Int.  s Thm. 3. Reflexive Property

Ex. 3: Proof of Theorem 6.8 Given: BC ║DA, BC ≅ DA Prove: ABCD is a  Statements: 1. BC ║DA 2.  DAC ≅  BCA  AC ≅ AC  BC ≅ DA 5. ∆BAC ≅ ∆DCA 6. AB ≅ CD  ABCD is a  Reasons: 1. Given 2. Alt. Int.  s Thm. 3. Reflexive Property 4. Given

Ex. 3: Proof of Theorem 6.8 Given: BC ║DA, BC ≅ DA Prove: ABCD is a  Statements: 1. BC ║DA 2.  DAC ≅  BCA  AC ≅ AC  BC ≅ DA 5. ∆BAC ≅ ∆DCA 6. AB ≅ CD  ABCD is a  Reasons: 1. Given 2. Alt. Int.  s Thm. 3. Reflexive Property 4. Given 5. SAS Congruence Post.

Ex. 3: Proof of Theorem 6.8 Given: BC ║DA, BC ≅ DA Prove: ABCD is a  Statements: 1. BC ║DA 2.  DAC ≅  BCA  AC ≅ AC  BC ≅ DA 5. ∆BAC ≅ ∆DCA 6. AB ≅ CD  ABCD is a  Reasons: 1. Given 2. Alt. Int.  s Thm. 3. Reflexive Property 4. Given 5. SAS Congruence Post. 6. CPCTC

Ex. 3: Proof of Theorem 6.8 Given: BC ║DA, BC ≅ DA Prove: ABCD is a  Statements: 1. BC ║DA 2.  DAC ≅  BCA  AC ≅ AC  BC ≅ DA 5. ∆BAC ≅ ∆DCA 6. AB ≅ CD  ABCD is a  Reasons: 1. Given 2. Alt. Int.  s Thm. 3. Reflexive Property 4. Given 5. SAS Congruence Post. 6. CPCTC 7. If opp. sides of a quad. are ≅, then it is a .

Assignment # 1-9, 12, 14-18