Which of the following conditions ensures that a quadrilateral is a parallelogram?
A) A pair of opposite sides are both congruent and parallel.
B) Both pairs of opposite sides are congruent.
C) A pair of opposite sides are congruent.
D) All of the above
http://en.wikipedia.org/wiki/Parallelogram
To determine which condition ensures that a quadrilateral is a parallelogram, we need to consider the properties of a parallelogram.
A parallelogram is a quadrilateral with two pairs of opposite sides that are both parallel and congruent. So, the correct condition would be the one that includes both parallel and congruent sides.
Let's analyze each option:
A) A pair of opposite sides are both congruent and parallel.
This option includes both the condition of congruent sides and parallel sides, which is a property of parallelograms. So, option A satisfies the condition for a parallelogram.
B) Both pairs of opposite sides are congruent.
While this option includes the condition of congruent sides, it does not specify anything about the sides being parallel, which is required for a parallelogram. Therefore, option B alone is not sufficient to ensure a quadrilateral is a parallelogram.
C) A pair of opposite sides are congruent.
Similar to option B, this only mentions congruent sides, but not parallel sides. Thus, option C alone is not sufficient to guarantee a parallelogram.
D) All of the above.
This option includes all the conditions mentioned in options A, B, and C. It states that a pair of opposite sides are both congruent and parallel, and that both pairs of opposite sides are congruent. Since it encompasses all the conditions for a parallelogram, option D satisfies the condition.
Therefore, the correct answer is D) All of the above, as it includes all the necessary conditions for a quadrilateral to be a parallelogram.