To prove that a quadrilateral is a parallelogram, we need to show that the opposite sides are congruent and that all angles are equal. This can be done by using any of the following methods:
1. Show that all angles are equal:
If we can show that all angles of the quadrilateral are equal, it implies that the quadrilateral is a rectangle, which is a special case of a parallelogram.
2. Show both pairs of opposite sides are congruent:
If we can show that both pairs of opposite sides are congruent, it implies that all sides of the quadrilateral are congruent, which makes it a parallelogram.
3. Show one pair of opposite sides are parallel AND congruent:
If we can show that one pair of opposite sides are parallel and congruent, it implies that the quadrilateral is a rhombus, which is a special case of a parallelogram.
4. Show both pairs of opposite sides are parallel:
If we can show that both pairs of opposite sides are parallel, it implies that the quadrilateral is a trapezoid, which is a special case of a parallelogram.
However, the method that is NOT a way to prove that a quadrilateral is a parallelogram is to show that one pair of opposite sides is not parallel. This method does not give us enough information about the quadrilateral, and does not guarantee that the quadrilateral is a parallelogram.