B Apply

15. Hardware You can use a simple device called a turnbuckle to “square up” structures that are parallelograms. For the gate pictured below, you tighten or loosen the turnbuckle on the diagonal cable so that the rectangular frame will keep the shape of a parallelogram when it sags. What are two ways you can make sure that the turnbuckle works? Explain.

    

16. Reasoning Suppose the diagonals of a parallelogram are both perpendicular and congruent. What type of special quadrilateral is it? Explain your reasoning.

Algebra For what value of x is the figure the given special parallelogram?

17. rectangle

    

18. rhombus

    

19. rectangle

    

Open-Ended Given two segments with lengths a and b ( a ≠ b ) , b open eh not equal to b close comma what special parallelograms meet the given conditions? Show each sketch.

20. Both diagonals have length a.
21. The two diagonals have lengths a and b.
22. One diagonal has length a, and one side of the quadrilateral has length b.

23. Proof Prove Theorem 6-17.

    Given: ABCD is a parallelogram.

    A C ¯ eh c bar bisects ∠ B A D angle b eh d and ∠ B C D . angle b c d .

    Prove: ABCD is a rhombus.

    

24. Proof Prove Theorem 6-18.

    Given: ▱ A B C D white parallelogram eh b c d , A C ¯ ≅ B D ¯ eh c bar , approximately equal to , b d bar

    Prove: ABCD is a rectangle.

    

Think About a Plan Explain how to construct each figure given its diagonals.

• What do you know about the diagonals of each figure?
• How can you apply constructions to what you know about the diagonals?

25. parallelogram
26. rectangle
27. rhombus

C Challenge

Determine whether the quadrilateral can be a parallelogram. Explain.

28. The diagonals are congruent, but the quadrilateral has no right angles.
29. Each diagonal is 3 cm long and two opposite sides are 2 cm long.
30. Two opposite angles are right angles, but the quadrilateral is not a rectangle.

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