Quadrilaterals

Parallelograms, trapezoids, and the special parallelograms.

The idea

A quadrilateral is a four-sided polygon, and the useful quadrilaterals are the families we get by adding conditions to that definition.

A parallelogram is a quadrilateral with both pairs of opposite sides parallel. From that single condition several properties follow: opposite sides are equal in length, opposite angles are equal, two angles sharing a side are supplementary (they add to $180°$, since they lie between the same pair of parallel lines), and the diagonals bisect each other.

One further condition gives the special parallelograms. Four right angles make a rectangle; four equal sides make a rhombus; a square is both at once. Every square is therefore a rectangle and a rhombus, while a rectangle need not have equal sides and a rhombus need not have right angles.

A trapezoid requires only one pair of parallel sides, its bases, so the parallelogram properties no longer apply. It has instead a median, the segment joining the midpoints of the two non-parallel sides; the median is parallel to the bases, and its length is their average, $\tfrac{b_1 + b_2}{2}$.

Ways to work on it

• Walkthrough. Work with the angle and side properties of parallelograms, trapezoids, and squares.
• Proof. See why a parallelogram's diagonals bisect each other.
• Practice. Find a missing angle in a parallelogram or the median of a trapezoid.
• Hardest. Solve for an unknown from a parallelogram's angle relationships.

Not sure where to start? Take the ten-question placement test.