Question

which statements prove that a quadrilateral is a parallelogram?
select each correct answer.

□ quadrilateral defg has opposite angles that are congruent

□ quadrilateral defg has one set of opposite sides that are both congruent and parallel.

□ quadrilateral defg has diagonals that are congruent.

□ quadrilateral defg has two sets of consecutive angles that are complementary

Explanation:

To determine which statements prove a quadrilateral is a parallelogram, we analyze each option:

1. Opposite angles congruent: A theorem states that if a quadrilateral has opposite angles congruent, it is a parallelogram. So this statement is valid.
2. One set of opposite sides congruent and parallel: A theorem confirms that if one pair of opposite sides of a quadrilateral is both congruent and parallel, the quadrilateral is a parallelogram. This is valid.
3. Diagonals congruent: Congruent diagonals are a property of rectangles (a type of parallelogram) but also hold for isosceles trapezoids (not parallelograms). Thus, this alone does not prove it is a parallelogram.
4. Two sets of consecutive angles complementary: Complementary angles sum to \(90^\circ\). For a parallelogram, consecutive angles are supplementary (sum to \(180^\circ\)), not complementary. This does not prove it is a parallelogram.

Answer:

• Quadrilateral \(DEFG\) has opposite angles that are congruent
• Quadrilateral \(DEFG\) has one set of opposite sides that are both congruent and parallel