Which statements prove that a quadrilateral is a parallelogram? select each answer

Let's explain each option:

A parallelogram is a quadrilateral where both pairs of opposite sides are parallel. On the other hand, the angles of every quadrilateral always add up to 360 degrees. Quadrilateral DEFG is shown in Figure 1 and the options tells us:

Quadrilateral DEFG has two sets of consecutive angles that are complementary.

So this is false because two angles are complementary if and only if they add up to 90°. If we take two consecutive angles, namely, angles d and e, they aren't complementary but supplementary for they add up to 180 degrees.

The statements tells us:

Quadrilateral DEFG has diagonals that are congruent

From the figure 1, the diagonals are the lines in blue and red. The diagonals in every parallelogram bisect each other but that doesn't mean it has to have the same length. This is only true if the parallelogram is a rectangle. Thus, if a parallelogram's diagonals have the same length, then it's definitely a rectangle.

The statements tells us:

Quadrilateral DEFG has opposite angles that are congruent

Opposite angles in parallelogram are always congruent. This can be prove by looking at Figure 2. Let's name a an angle of the parallelogram. So angle a and b are alternate interior angles, therefore a = b. So the supplementary angle to b is 180° - b or 180° - a. Also, b is corresponding to c, hence they are the same and a = b = c. Finally, opposite angles in parallelogram are always congruent.

The statements tells us:

Quadrilateral DEFG has one set of opposite sides that are both congruent and parallels

The definition of parallelograms tells us that a parallelogram is a quadrilateral where both pairs of opposite sides are parallel. Therefore, the second part of the statement is true. On the other hand, opposite sides in parallelograms are always congruent. So the entire statement is true.