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Properties and Proofs of Parallelograms

Learn about the conditions for parallelograms and how to prove that a quadrilateral is a parallelogram. Explore examples using geometry and algebra, as well as coordinate proofs.

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Properties and Proofs of Parallelograms

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  1. Splash Screen

  2. Five-Minute Check (over Lesson 6–2) Then/Now Theorems: Conditions for Parallelograms Proof: Theorem 6.9 Example 1: Identify Parallelograms Example 2: Real-World Example: Use Parallelograms to Prove Relationships Example 3: Use Parallelograms and Algebra to Find Values Concept Summary: Prove that a Quadrilateral Is a Parallelogram Example 4: Parallelograms and Coordinate Geometry Example 5: Parallelograms and Coordinate Proofs Lesson Menu

  3. A B C ? ____ A. B. C. 5-Minute Check 1

  4. A B C ? A. B. C. 5-Minute Check 2

  5. A B C ? A. A B. B C. C 5-Minute Check 3

  6. A B C D A.A  C and B  D B.A  B and C  D C. D. An expandable gate is made of parallelograms that have angles that change measure as the gate is adjusted. Which of the following statements about the angles in the gate is always true? 5-Minute Check 4

  7. You recognized and applied properties of parallelograms. (Lesson 6–2) • Recognize the conditions that ensure a quadrilateral is a parallelogram. • Prove that a set of points forms a parallelogram in the coordinate plane. Then/Now

  8. Concept 1

  9. Concept 2

  10. Identify 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. Example 1

  11. A B C D 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 . Example 1

  12. Use Parallelograms to Prove Relationships MECHANICS Scissor lifts, like the platform lift shown below, are commonly applied to tools intended to lift heavy items. In the diagram, A  C and B  D. Explain why the consecutive angles will always be supplementary, regardless of the height of the platform. Example 2

  13. Use Parallelograms to Prove Relationships Answer: Since both pairs of opposite angles of quadrilateral ABCD are congruent, ABCD is a parallelogram by Theorem 6.10. Theorem 6.5 states that consecutive angles of parallelograms are supplementary. Therefore, mA + mB = 180 and mC + mD = 180. By substitution, mA + mD = 180 and mC + mB = 180. Example 2

  14. A B C D The diagram shows a car jack used to raise a car from the ground. In the diagram, AD BC and AB  DC. Based on this information, which statement will be true, regardless of the height of the car jack. A. A  B B. A  C C.AB  BC D.mA + mC = 180 Example 2

  15. Use Parallelograms and Algebra to Find Values Find x and y so that the quadrilateral is a parallelogram. Opposite sides of a parallelogram are congruent. Example 3

  16. Use Parallelograms and Algebra to Find Values AB = DC Substitution Distributive Property Subtract 3x from each side. Add 1 to each side. Example 3

  17. Use Parallelograms and Algebra to Find Values Substitution Distributive Property Subtract 3y from each side. Add 2 to each side. Answer: So, when x = 7 and y = 5, quadrilateral ABCD is a parallelogram. Example 3

  18. A B C D Find m so that the quadrilateral is a parallelogram. A.m = 2 B.m = 3 C.m = 6 D.m = 8 Example 3

  19. Concept 3

  20. Parallelograms and Coordinate Geometry COORDINATE GEOMETRYGraph quadrilateral QRST with vertices Q(–1, 3), R(3, 1), S(2, –3), and T(–2, –1). Determine whether the quadrilateral is a parallelogram. Justify your answer by using the Slope Formula. If the opposite sides of a quadrilateral are parallel, then it is a parallelogram. Example 4

  21. Answer: Since opposite sides have the same slope, QR║ST and RS║TQ. Therefore, QRST is a parallelogram by definition. Parallelograms and Coordinate Geometry Example 4

  22. A B Graph quadrilateral EFGH with vertices E(–2, 2), F(2, 0), G(1, –5), and H(–3, –2). Determine whether the quadrilateral is a parallelogram. A. yes B. no Example 4

  23. Step 1 Position quadrilateral ABCD on the coordinate plane such that AB DC and AD  BC. ● Let AB have a length of a units. Then B has coordinates (a, 0). Parallelograms and Coordinate Proofs Write a coordinate proof for the following statement. If both pairs of opposite sides of a quadrilateral are congruent, then the quadrilateral is a parallelogram. ● Begin by placing the vertex A at the origin. Example 5

  24. ● Since AD BC position the endpoints of DC so that they have the same y-coordinate, c. Parallelograms and Coordinate Proofs ● So that the distance from D to C is also a units, let the x-coordinate of D be b and of C be b + a. Example 5

  25. Given: quadrilateral ABCD, AB DC, AD  BC Parallelograms and Coordinate Proofs Step 2 Use your figure to write a proof. Prove:ABCD is a parallelogram. Coordinate Proof: By definition a quadrilateral is a parallelogram if opposite sides are parallel. Use the Slope Formula. Example 5

  26. The slope of AB is 0. The slope of CD is 0. Since AB and CD have the same slope and AD and BC have the same slope, AD║BC and AB║CD. Parallelograms and Coordinate Proofs Answer: So, quadrilateral ABCD is a parallelogram because opposite sides are parallel. Example 5

  27. A B A.AB = a units and DC = a units; slope of AB = 0 and slope of DC = 0 B.AD = c units and BC = c units; slope of and slope of Which of the following can be used to prove the statement below? If a quadrilateral is a parallelogram, then one pair of opposite sides is both parallel and congruent. Example 5

  28. End of the Lesson

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