Question 251824: A parallelogram is sometimes 1. equangular if it is equilateral.
2. equilateral if it is equiangular.
correct?
Also, diagonals of a kite are always perpendicular bisectors of one another. correct?
The diagonals of a trapezoid sometimes bisect one another. correct?
Answer by solver91311(24713)   (Show Source):
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1. Yes. A rhombus is a parallelogram that is equilateral but not equiangular, but a square is a special case of a parallelogram that is both equiangular and equilateral and this is the only case that qualifies your "sometimes."
2. Yes. A rectangle is a parallelogram that is equiangular but not equilateral, but a square is a special case of a parallelogram that is both equiangular and equilateral and this is the only case that qualifies your "sometimes."
3. No. The diagonals of a kite are always perpendicular, but only in the special case where a kite is also a square are they bisectors.
4. This depends on your definition of a trapezoid.
Definition 1: A trapezoid is a quadrilateral with exactly one pair of parallel sides.
Definition 2: A trapezoid is a quadrilateral with at least one pair of parallel sides.
In general, the diagonals of a trapezoid divide each other in the same ratio as the ratio of the lengths of the unequal sides. So, in order for the diagonals to bisect each other, the measures of the "unequal" sides would have to be equal. This is only possible if the quadrilateral has two pairs of parallel sides, as is possible under definition 2 which allows a parallelogram to be a special case of the trapezoid.
In sum, if definition 1, the answer to #4 is No. If definition 2, the answer is yes, but only in the special case of a parallelogram. Some authorities use one of these definitions and others use the other. Read the definition in your text and discuss it with your instructor.
John
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