A quadrilateral is NOT a parallelogram if and only if
A. the diagonals bisect each other.
B. the opposite angles are congruent.
C. two consecutive angles are supplementary.
D. the non-opposite sides are parallel and congruent.
The correct answer and explanation is:
The correct answer is C. two consecutive angles are supplementary.
A parallelogram is defined as a quadrilateral in which both pairs of opposite sides are parallel. In a parallelogram, certain properties hold true, such as opposite sides being equal in length, opposite angles being congruent, and the diagonals bisecting each other.
However, if a quadrilateral is not a parallelogram, one of the properties may fail. Specifically, in a non-parallelogram quadrilateral, two consecutive angles are supplementary, meaning that their sum is 180 degrees. This occurs because the consecutive angles form a linear pair, and this property is not exclusive to parallelograms.
The other options refer to properties that are characteristic of parallelograms:
- A. The diagonals bisect each other: This is a defining property of parallelograms, so if a quadrilateral does not have this property, it is not a parallelogram.
- B. The opposite angles are congruent: In a parallelogram, opposite angles are always congruent. A quadrilateral that does not have congruent opposite angles cannot be a parallelogram.
- D. The non-opposite sides are parallel and congruent: This is a feature of rectangles, squares, and rhombuses, but not a general property of parallelograms. Parallelograms may have opposite sides that are parallel and congruent, but the non-opposite sides being both parallel and congruent would make it a special type of parallelogram, like a rhombus.
Therefore, the key feature that distinguishes a non-parallelogram from a parallelogram is the fact that two consecutive angles are supplementary, as stated in option C.