Parallelograms
A parallelogram has two pairs of parallel sides. That single condition gives it a whole family of properties, and several ways to prove a shape is one.
Definition
A parallelogram is a quadrilateral with both pairs of opposite sides parallel. It is written with the symbol of a slanted box followed by its vertices in order, such as parallelogram ABCD, where AB is parallel to CD and BC is parallel to DA.
Sides and angles
In a parallelogram, opposite sides are congruent and opposite angles are congruent. Consecutive angles, sharing a side, are supplementary, because they are same-side interior angles between parallel lines. If one angle is 70 degrees, the angles are 70, 110, 70, and 110.
Diagonals
The diagonals of a parallelogram bisect each other: they cross at their midpoints. If one diagonal is split into pieces 3x + 1 and 2x + 5, then 3x + 1 = 2x + 5, so x = 4 and each piece is 13. The diagonals are not generally equal in length.
Why the properties hold
Drawing one diagonal splits a parallelogram into two triangles that are congruent by ASA, using the alternate interior angles formed by the diagonal crossing each pair of parallel sides. CPCTC then gives the congruent opposite sides and angles.
Proving a parallelogram
A quadrilateral is a parallelogram if any of these hold: both pairs of opposite sides are parallel; both pairs of opposite sides are congruent; both pairs of opposite angles are congruent; the diagonals bisect each other; or one pair of opposite sides is both congruent and parallel.
Area of a parallelogram
The area of a parallelogram is base times height, \(A = bh\), where the height is perpendicular to the base, not the slanted side. Cutting a triangle off one end and moving it to the other turns the parallelogram into a rectangle with the same base and height.
Parallelograms in mechanisms
Because opposite sides stay parallel as the angles change, parallelogram linkages keep objects level while they move: desk lamps, scissor lifts, car-park barriers, and the pantograph, a drawing tool for copying and enlarging, all use them.
Worked example
In parallelogram PQRS, angle P measures 2x + 20 degrees and angle Q measures 3x - 15 degrees. Consecutive angles are supplementary, so 5x + 5 = 180, giving x = 35. Angle P is 90 degrees and so is angle Q, so this parallelogram is in fact a rectangle.
Key takeaways
A parallelogram has both pairs of opposite sides parallel. Its opposite sides and opposite angles are congruent, consecutive angles are supplementary, and its diagonals bisect each other. These properties follow from two congruent triangles formed by a diagonal. A quadrilateral is a parallelogram if its opposite sides are parallel or congruent, its opposite angles are congruent, its diagonals bisect each other, or one pair of sides is both parallel and congruent. Its area is base times perpendicular height.
Flashcard
Both pairs of opposite sides parallel
Check your understanding
One angle of a parallelogram is 70 degrees. What are the other angles?