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8.3 Tests for Parallelograms

8.3 Tests for Parallelograms. Objectives. Recognize the conditions that ensure a quadrilateral is a parallelogram. Prove that a set of points forms a parallelogram in the coordinate plane. Conditions for a Parallelogram.

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8.3 Tests for Parallelograms

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  1. 8.3 Tests for Parallelograms

  2. Objectives • Recognize the conditions that ensure a quadrilateral is a parallelogram. • Prove that a set of points forms a parallelogram in the coordinate plane.

  3. Conditions for a Parallelogram • Obviously, if the opposite sides of a quadrilateral are parallel, then it is a parallelogram; but there are other tests we can also apply to a quadrilateral to test whether it is a parallelogram or not.

  4. Conditions for a Theorems • Theorem 8.9 – If both pairs of opposite sides are ≅, then the quad. is a . • Theorem 8.10 – If both pairs of opposite s are ≅, then the quad. is a . • Theorem 8.11 – If diagonals bisect each other, then the quad. is . • Theorem 8.12 – If one pair of opposite sides is ║ and ≅, then the quad. is a .

  5. Given: Proof: CPCTC. By Theorem 8.9, if both pairs of opposite sides of a quadrilateral are congruent, the quadrilateral is a parallelogram. Therefore, ABCD is a parallelogram. Example 1: Write a paragraph proof of the statement: If a diagonal of a quadrilateral divides the quadrilateral into two congruent triangles, then the quadrilateral is a parallelogram. Prove:ABCD is a parallelogram.

  6. Given: Your Turn: Write a paragraph proof of the statement: If two diagonals of a quadrilateral divide the quadrilateral into four triangles where opposite triangles are congruent, then the quadrilateral is a parallelogram. Prove:WXYZ is a parallelogram.

  7. Proof: by CPCTC. By Theorem 8.9, if both pairs of opposite sides of a quadrilateral are congruent, the quadrilateral is a parallelogram. Therefore, WXYZ is a parallelogram. Your Turn:

  8. Example 2: Some of the shapes in this Bavarian crest appear to be parallelograms. Describe the information needed to determine whether the shapes are parallelograms. Answer: If both pairs of opposite sides are the same length or if one pair of opposite sides is a congruent and parallel, the quadrilateral is a parallelogram. If both pairs of opposite angles are congruent or if the diagonals bisect each other, the quadrilateral is a parallelogram.

  9. Your Turn: The shapes in the vest pictured here appear to be parallelograms. Describe the information needed to determine whether the shapes are parallelograms. Answer: If both pairs of opposite sides are the same length or if one pair of opposite sides is congruent and parallel, the quadrilateral is a parallelogram. If both pairs of opposite angles are congruent or if the diagonals bisect each other, the quadrilateral is a parallelogram.

  10. Example 3: Determine whether the quadrilateral is a parallelogram. Justify your answer. Answer: Each pair of opposite sides have the same measure. Therefore, they are congruent. If both pairs of opposite sides of a quadrilateral are congruent, the quadrilateral is a parallelogram.

  11. Your Turn: Determine whether the quadrilateral is a parallelogram. Justify your answer. Answer: One pair of opposite sides is parallel and has the same measure, which means these sides are congruent. If one pair of opposite sides of a quadrilateral is both parallel and congruent, then the quadrilateral is a parallelogram.

  12. Tests for Parallelograms • Both pairs of opposite sides are parallel. • Both pairs of opposite sides are congruent. • Both pairs of opposite angles are congruent. • The diagonals bisect each other. • A pair of opposite sides is both parallel and congruent.

  13. A B D C Example 4a: Find x so that the quadrilateral is a parallelogram. Opposite sides of a parallelogram are congruent.

  14. Example 4a: Substitution Distributive Property Subtract 3x from each side. Add 1 to each side. Answer: When x is 7, ABCD is a parallelogram.

  15. D E G F Example 4b: Find y so that the quadrilateral is a parallelogram. Opposite angles of a parallelogram are congruent.

  16. Example 4b: Substitution Subtract 6y from each side. Subtract 28 from each side. Divide each side by –1. Answer: DEFG is a parallelogram when y is 14.

  17. a. b. Answer: Answer: Your Turn: Find m and n so that each quadrilateral is a parallelogram.

  18. Parallelograms on the Coordinate Plane • We can use the Distance Formula and the Slope Formula to determine if a quadrilateral is a parallelogram on the coordinate plane. • Just pick one of the tests… and apply either or both of the formulas.

  19. Example 5a: 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.

  20. Answer: Since opposite sides have the same slope, Therefore, ABCD is a parallelogram by definition. Example 5a: If the opposite sides of a quadrilateral are parallel, then it is a parallelogram.

  21. Example 5b: COORDINATE GEOMETRYDetermine whether the figure with vertices P(–3, –1), Q(–1, 3), R(3, 1), and S(1, –3)is a parallelogram. Use the Distance and Slope Formulas.

  22. Example 5b: First use the Distance Formula to determine whether the opposite sides are congruent.

  23. Next, use the Slope Formula to determine whether and have the same slope, so they are parallel. Example 5b: Answer: Since one pair of opposite sides is congruent and parallel, PQRS is a parallelogram.

  24. Your Turn: Determine whether the figure with the given vertices is a parallelogram. Use the method indicated. a. A(–1, –2), B(–3, 1), C(1, 2), D(3, –1); Slope Formula

  25. Answer: The slopes of and the slopes of Therefore, Since opposite sides are parallel, ABCD is a parallelogram. Your Turn:

  26. Distance and Slope Formulas b. L(–6, –1), M(–1, 2), N(4, 1), O(–1, –2); Your Turn: Determine whether the figure with the given vertices is a parallelogram. Use the method indicated.

  27. Answer: Since the slopes of Since one pair of opposite sides is congruent and parallel, LMNO is a parallelogram. Your Turn:

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