Question

Which statements prove that a quadrilateral is a parallelogram? Select each correct answer.

A) Quadrilateral DEFG has diagonals that bisect each other.
B) Quadrilateral D E F G, has diagonals that bisect each other.
C) Quadrilateral DEFG has only one set of opposite sides that are congruent.
D) Quadrilateral D E F G, has only one set of opposite sides that are congruent.
E) Quadrilateral DEFG has opposite angles that are congruent.
F) Quadrilateral D E F G, has opposite angles that are congruent.
G) Quadrilateral DEFG has only one set of consecutive angles that are supplementary.

Answer 1

To prove a quadrilateral is a parallelogram, the diagonals must bisect each other, or the opposite angles must be congruent. Only options A, B, E, and F describe these properties correctly.

Step-by-step explanation:

To determine if a quadrilateral is a parallelogram, one must analyze its properties and apply certain theorems and postulates. There are a few key characteristics that prove a quadrilateral is a parallelogram:

• If a quadrilateral's opposite sides are congruent, it's a parallelogram. However, having only one set of opposite sides congruent is not enough to prove it's a parallelogram, which eliminates options C and D.
• If the diagonals of a quadrilateral bisect each other, this is a strong indicator that the shape is a parallelogram, which makes options A and B correct.
• Having opposite angles that are congruent also proves that a quadrilateral is a parallelogram, which confirms options E and F as correct.
• A quadrilateral where only one set of consecutive angles is supplementary does not necessarily make it a parallelogram, ruling out option G.

Therefore, the correct statements proving that a quadrilateral is a parallelogram are that it has diagonals that bisect each other (A and B) and opposite angles that are congruent (E and F).