Question

9 quadrilateral abcd is a parallelogram because (1) two of the angles have equal measures. (2) two of the opposite sides are congruent.. (3) two of the opposite sides are parallel. (4) two pairs of opposite angles are congruent

Explanation:

To determine why quadrilateral \(ABCD\) is a parallelogram, we analyze each option:

• Option (1): Having two angles with equal measures is not a sufficient condition for a parallelogram (e.g., an isosceles trapezoid has equal base angles but is not a parallelogram).
• Option (2): Two pairs of opposite sides being congruent is a parallelogram property, but "two of the opposite sides" (implying only one pair) is insufficient (e.g., an isosceles trapezoid has one pair of congruent sides).
• Option (3): Two pairs of opposite sides being parallel is a definition, but "two of the opposite sides" (one pair) is insufficient (e.g., a trapezoid has one pair of parallel sides).
• Option (4): A key theorem states that if a quadrilateral has two pairs of opposite angles congruent, it is a parallelogram. This satisfies the parallelogram criterion.

Answer:

(4) two pairs of opposite angles are congruent