5.6 Proving Quadrilaterals are Parallelograms

Objectives: Prove that a quadrilateral is a parallelogram.

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

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

Theorems Theorem: If an angle of a quadrilateral is supplementary to both of its consecutive angles, then the quadrilateral is a parallelogram. ABCD is a parallelogram. x°x° (180 – x) ° x°x°

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

Ex. 1: Proof of Theorem 1 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 1 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 1 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 1 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 1 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 1 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.

Ex. 2: Proving Quadrilaterals are Parallelograms As the sewing box below is opened, the trays are always parallel to each other. Why?

Ex. 2: Proving Quadrilaterals are Parallelograms Each pair of hinges are opposite sides of a quadrilateral. The 2.75 inch sides of the quadrilateral are opposite and congruent. The 2 inch sides are also opposite and congruent. Because opposite sides of the quadrilateral are congruent, it is a parallelogram. By the definition of a parallelogram, opposite sides are parallel, so the trays of the sewing box are always parallel.

Another Theorem ~ Theorem—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 Given: BC ║DA, BC ≅ DA Prove: ABCD is a  Statements: 1. BC ║DA 2.  DAC ≅  BCA 3. AC ≅ AC 4. BC ≅ DA 5. ∆BAC ≅ ∆DCA 6. AB ≅ CD 7. ABCD is a  Reasons: 1. Given

Ex. 3: Proof of Theorem Given: BC ║DA, BC ≅ DA Prove: ABCD is a  Statements: 1. BC ║DA 2.  DAC ≅  BCA 3. AC ≅ AC 4. BC ≅ DA 5. ∆BAC ≅ ∆DCA 6. AB ≅ CD 7. ABCD is a  Reasons: 1. Given 2. Alt. Int.  s Thm.

Ex. 3: Proof of Theorem Given: BC ║DA, BC ≅ DA Prove: ABCD is a  Statements: 1. BC ║DA 2.  DAC ≅  BCA 3. AC ≅ AC 4. BC ≅ DA 5. ∆BAC ≅ ∆DCA 6. AB ≅ CD 7. ABCD is a  Reasons: 1. Given 2. Alt. Int.  s Thm. 3. Reflexive Property

Ex. 3: Proof of Theorem Given: BC ║DA, BC ≅ DA Prove: ABCD is a  Statements: 1. BC ║DA 2.  DAC ≅  BCA 3. AC ≅ AC 4. BC ≅ DA 5. ∆BAC ≅ ∆DCA 6. AB ≅ CD 7. ABCD is a  Reasons: 1. Given 2. Alt. Int.  s Thm. 3. Reflexive Property 4. Given

Ex. 3: Proof of Theorem Given: BC ║DA, BC ≅ DA Prove: ABCD is a  Statements: 1. BC ║DA 2.  DAC ≅  BCA 3. AC ≅ AC 4. BC ≅ DA 5. ∆BAC ≅ ∆DCA 6. AB ≅ CD 7. 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 Given: BC ║DA, BC ≅ DA Prove: ABCD is a  Statements: 1. BC ║DA 2.  DAC ≅  BCA 3. AC ≅ AC 4. BC ≅ DA 5. ∆BAC ≅ ∆DCA 6. AB ≅ CD 7. 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 Given: BC ║DA, BC ≅ DA Prove: ABCD is a  Statements: 1. BC ║DA 2.  DAC ≅  BCA 3. AC ≅ AC 4. BC ≅ DA 5. ∆BAC ≅ ∆DCA 6. AB ≅ CD 7. 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 .