Choose from the following statements the properties of a parallelogram: A) it has two pairs of opposite sides of equal length, B) it is a quadrilateral of four equal angles, C) it has one pair of perpendicular sides, D) it has four sides of equal length, E) it has two pairs of equal angles.
Let’s write a definition of what we know to be true of a parallelogram. A parallelogram is a four-sided flat shape with straight sides, where opposite sides are parallel, opposite sides are equal in length, and opposite angles are equal.
Based on this definition, let’s walk back through statements A through E to see which ones would be true. It has two pairs of opposite sides of equal length. Yes, that is true of a parallelogram. It is a quadrilateral of four equal angles. It is four-sided. So it is a quadrilateral. But opposite angles are equal. And sometimes that will mean all four angles are equal, for example, in a square.
But that’s not always the case. So we can’t say that that is true. We can only say opposite angles are equal, not that all four angles are equal. It has one pair of perpendicular sides. Well, some parallelograms may have perpendicular sides. It’s not a requirement.
Statement D, it has four sides of equal length. We know that can’t be true. A rectangle is a parallelogram. So it doesn’t have to have four sides of equal length to be a parallelogram. Opposite sides are equal in length, but not all four sides.
So far, we still only have statement A that is always true. Final statement, statement E, it has two pairs of equal angles. This is true. Opposite angles are equal. And because it’s a quadrilateral, there will be two sets of angles. Opposite angles are equal. Two pairs of equal angles, that’s true. So we can say that statement A and statement E are true for all parallelograms.