Question

To prove that a given quadrilateralIs a parallelogram it is enough to show that

a. Opposite angles are congruent
b. Diagonals bisect each other
c. All angles are right angles
d. Consecutive angles are supplementary

Answer 1

To prove a quadrilateral is a parallelogram, showing that the diagonals bisect each other is an effective method. Although other properties can also indicate a parallelogram, this approach provides a direct geometric proof.

Step-by-step explanation:

To prove that a given quadrilateral is a parallelogram, one effective method is to show that the diagonals bisect each other. If the diagonals of a quadrilateral intersect at their midpoints, this indicates that both pairs of opposite sides are parallel and equal in length, satisfying the definition of a parallelogram. The options provided including 'opposite angles are congruent', 'diagonals bisect each other', 'all angles are right angles', and 'consecutive angles are supplementary' are all characteristics that could indicate a quadrilateral is a parallelogram, but showing the diagonals bisect each other is a direct approach with clear geometric proof.

Keep in mind the other properties as well for a comprehensive understanding. For instance, if all angles in a quadrilateral are right angles, then it is a rectangle, which is a special type of parallelogram. Similarly, if consecutive angles are supplementary, it supports the fact that the figure could be a parallelogram as the opposite sides are parallel. Remember that these properties might apply to specific cases or types of parallelograms.

Answer 2

To prove that a quadrilateral is a parallelogram, demonstrating that diagonals bisect each other is sufficient. This criterion, along with showing that consecutive angles are supplementary, are properties of parallelograms.

Step-by-step explanation:

The question pertains to the properties that indicate a quadrilateral is a parallelogram. In mathematics, particularly in geometry, several criteria can be used to prove that a quadrilateral is a parallelogram. Out of the options provided, showing that the diagonals bisect each other is one of the properties that confirm the given quadrilateral is indeed a parallelogram. This is because in a parallelogram, each diagonal divides the figure into two congruent triangles, thereby ensuring that the diagonals bisect each other at their point of intersection.

It is worth noting that showing opposite angles to be congruent (option a) or that all angles are right angles (option c), although they may occur in certain parallelograms like rectangles, are not definitive proofs for all parallelograms. However, option d, stating that consecutive angles are supplementary, can also be a valid proof as it is a property consistent with parallelograms.