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Proving Quadrilaterals are // ograms .

Learn how to prove that a quadrilateral is a parallelogram using various methods and properties such as opposite sides and angles, diagonals, and congruent sides.

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Proving Quadrilaterals are // ograms .

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  1. Proving Quadrilaterals are //ograms. Chapter 6.3

  2. Proving Quadrilaterals are //ograms.Methods 1-3 • If opposite sides of a quadrilateral are //, then it is a //ogram. (definition) • If both pairs of opposite sides of a quadrilateral are , then it is a //ogram. • If both pairs of opposite angles are , then it is a //ogram.

  3. Proving Quadrilaterals are //ograms.Methods 4-6 Additional Test for a //ogram • If an angle of a quadrilateral is supplementary to both of its consecutive angles, then it is a //ogram. • If the diagonals bisect each other, then it is a parallelogram. • If one pair of opposite sides of a quadrilateral are // and , then it is a parallelogram. (new)

  4. Is this a Parallelogram? Why? Yes. Opposite Angles are Congruent.

  5. Is this a Parallelogram? Why? No, not enough information.

  6. Is this a Parallelogram? Why? Yes. Opposite Sides are Parallel (definition).

  7. Is this a Parallelogram? Why? Yes. One pair of opposite sides are parallel and congruent.

  8. Is this a Parallelogram? Why? 120o 60o 120o Yes. An angle is supplementary to both of its consecutive angles.

  9. Is this a Parallelogram? Why? No, not enough information.

  10. Is this a Parallelogram? Why? Yes. Opposite Sides are Congruent.

  11. Is this a Parallelogram? Why? 120o 60o No, not enough information.

  12. Is this a Parallelogram? Why? Yes. Diagonals bisect each other.

  13. Is this a Parallelogram? Why? No, not enough information.

  14. Is this a Parallelogram? Why? A B ABC CDA D C Yes, Opposite sides are congruent. Others can be proven as well.

  15. Reminders: Coordinate Proofs • Midpoint Formula • Slope// lines have equal slope Distance Formula

  16. Prove this is a Parallelogram

  17. Prove this is a Parallelogram • Slope & Distance • Prove AB = CD and AB // CD • Use Distance Formula to show that their lengths are equal and use slope formula to show that their slopes are equal. • Midpoint Method • Prove the diagonals bisect each other • Show that the diagonals have the same midpoint. Slope Method • Prove AB//CD and BC//AD • Use slope formula and show that their slopes are equal. Distance Method • Prove AB = CD and BC = AD • Use Distance Formula to show that their lengths are equal.

  18. Proof:Since ΔXVYΔZVW and ΔXVWΔZVY, by CPCTC. By which method would you prove WXYZ is a parallelogram? A. Both pairs of opp. sides . B. Both pairs of opp. ’s . C. One pair of opp. sides both  and ||. D. Diagonals bisect each other

  19. Properties of Parallelograms Determine whether the quadrilateral is a parallelogram. Justify your answer. Answer: Each pair of opposite sides has the same measure. Therefore, they are congruent. If both pairs of opposite sides of a quadrilateral are congruent, the quadrilateral is a parallelogram.

  20. Which method would prove the quadrilateral is a parallelogram? A. Both pairs of opp. sides ||. B. Both pairs of opp. sides . C. Both pairs of opp. ’s . D. One pair of opp. sides both || and .

  21. Find x so that the quadrilateral is a //ogram. Opposite sides of a //ogram are congruent.

  22. Find m so that the quadrilateral is a //ogram. A.m = 2 B.m = 3 C.m = 6 D.m = 8

  23. Use Slope and Distance COORDINATE GEOMETRYDetermine whether the figure with vertices A(–3, 0), B(–1, 3), C(3, 2), and D(1, –1) is a parallelogram. Use the Slope Formula.

  24. = slopes  // Lines Opp. Sides are //  //ogram

  25. Determine whether the figure with the given vertices is a parallelogram. Use the method indicated. A(–1, –2), B(–3, 1), C(1, 2), D(3, –1); Slope Formula A. yes B. no C. cannot be determined • A • B • C

  26. Homework Chapter 6.3 • Pg 337: 3-14, 20-25, 26, 28, 45-48

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