6-5 Conditions for Rhombuses, Rectangles, and Squares

Quick Review

If one diagonal of a parallelogram bisects two angles of the parallelogram, then the parallelogram is a rhombus. If the diagonals of a parallelogram are perpendicular, then the parallelogram is a rhombus. If the diagonals of a parallelogram are congruent, then the parallelogram is a rectangle.

Example

Can you conclude that the parallelogram is a rhombus, rectangle, or square? Explain.

Yes, the diagonals are perpendicular, so the parallelogram is a rhombus.

Exercises

Can you conclude that the parallelogram is a rhombus, rectangle, or square? Explain.

For what value of x is the figure the given parallelogram? Justify your answer.

31. 

    Rhombus

32. 

    Rectangle

6-6 Trapezoids and Kites

Quick Review

The parallel sides of a trapezoid are its bases and the nonparallel sides are its legs. Two angles that share a base of a trapezoid are base angles of the trapezoid. The midsegment of a trapezoid joins the midpoints of its legs.

The base angles of an isosceles trapezoid are congruent. The diagonals of an isosceles trapezoid are congruent.

The diagonals of a kite are perpendicular.

Example

ABCD is an isosceles trapezoid. What is m ∠ C ? m angle c question mark

Since B C ¯ ∥ A D ¯ , ∠ C b c bar , parallel to , eh d bar , comma . angle c  and ∠ D angle d  are same-side interior angles.

m ∠ C + m ∠ D = 180 Same-side interior angles are supplementary. m ∠ C + 60 = 180 Substitute. m ∠ C = 120 Subtract  60  from each side. table with 3 rows and 3 columns , row1 column 1 , m angle c plus m angle d , column 2 equals 180 , column 3 cap sameminussideinterioranglesaresupplementary. , row2 column 1 , m angle , c plus , 60 , column 2 equals 180 , column 3 cap substitute. , row3 column 1 , m angle c , column 2 equals 120 , column 3 cap subtract . 60 . fromeachside. , end table

Exercises

Find the measures of the numbered angles in each isosceles trapezoid.

Find the measures of the numbered angles in each kite.

35. 
36. 
    Image Long Description
37. Algebra A trapezoid has base lengths of ( 6 x − 1 ) open 6 x minus 1 close  units and 3 units. Its midsegment has a length of ( 5 x − 3 ) open 5 x minus 3 close  units. What is the value of x?

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