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8.3 Proving Quadrilaterals are Parallograms

8.3 Proving Quadrilaterals are Parallograms. 8.3 Proving Quadrilaterals are Parallelograms. Geometry. Objectives:. Prove that a quadrilateral is a parallelogram. Use coordinate geometry with parallelograms.

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8.3 Proving Quadrilaterals are Parallograms

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  1. 8.3Proving Quadrilaterals are Parallograms

  2. 8.3 Proving Quadrilaterals are Parallelograms Geometry

  3. Objectives: • Prove that a quadrilateral is a parallelogram. • Use coordinate geometry with parallelograms.

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

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

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

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

  8. 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

  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 Ex. 1: Proof of Theorem

  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 Ex. 1: Proof of Theorem

  11. 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

  12. 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

  13. 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

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

  15. 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. Ex. 2: Proving Quadrilaterals are Parallelograms

  16. 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. B C A D

  17. Objective 2: Using Coordinate Geometry • When a figure is in the coordinate plane, you can use the Distance Formula (see—it never goes away) to prove that sides are congruent and you can use the slope formula (see how you use this again?) to prove sides are parallel.

  18. Show that A(2, -1), B(1, 3), C(6, 5) and D(7,1) are the vertices of a parallelogram. Ex. 4: Using properties of parallelograms

  19. Method 1—Show that opposite sides have the same slope, so they are parallel. Slope of AB. 3-(-1) = - 4 1 - 2 Slope of CD. 1 – 5 = - 4 7 – 6 Slope of BC. 5 – 3 = 2 6 - 1 5 Slope of DA. - 1 – 1 = 2 2 - 7 5 AB and CD have the same slope, so they are parallel. Similarly, BC ║ DA. Ex. 4: Using properties of parallelograms Because opposite sides are parallel, ABCD is a parallelogram.

  20. Method 2—Show that opposite sides have the same length. AB=√(1 – 2)2 + [3 – (- 1)2] = √17 CD=√(7 – 6)2 + (1 - 5)2 = √17 BC=√(6 – 1)2 + (5 - 3)2 = √29 DA= √(2 – 7)2 + (-1 - 1)2 = √29 AB ≅ CD and BC ≅ DA. Because both pairs of opposites sides are congruent, ABCD is a parallelogram. Ex. 4: Using properties of parallelograms

  21. Method 3—Show that one pair of opposite sides is congruent and parallel. Slope of AB = Slope of CD = -4 AB=CD = √17 AB and CD are congruent and parallel, so ABCD is a parallelogram. Ex. 4: Using properties of parallelograms

  22. Classwork/Homework:

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