Rectangles, rhombuses, squares, and trapezoids
Special quadrilaterals add extra conditions to the parallelogram. Each extra condition brings extra properties, especially for the diagonals.
Rectangles
A rectangle is a parallelogram with four right angles. Besides all parallelogram properties, its diagonals are congruent. Carpenters check that a frame is rectangular by measuring both diagonals: if they are equal, the corners are square.
Rhombuses
A rhombus is a parallelogram with four congruent sides. Its diagonals are perpendicular, and each diagonal bisects a pair of opposite angles. The area of a rhombus is half the product of its diagonals: diagonals of 10 and 24 give an area of 120, and each side is the square root of (5 squared + 12 squared) = 13.
Squares
A square is both a rectangle and a rhombus: four right angles and four congruent sides. It has every property of both, so its diagonals are congruent, perpendicular, bisect each other, and bisect its angles. A square with side s has diagonals of s times root 2.
The family tree
Every square is a rhombus and a rectangle; every rhombus and rectangle is a parallelogram; and every parallelogram is a quadrilateral. The reverse statements are false: a rectangle need not be a square. Thinking of definitions as nested sets avoids confusion.
Trapezoids
A trapezoid has exactly one pair of parallel sides, called bases; the other two sides are legs. Some textbooks define it as at least one pair, which makes parallelograms trapezoids too. Its midsegment, joining the midpoints of the legs, is parallel to the bases and equal to their average, and its area is \(\tfrac{1}{2}(b_1 + b_2)h\).
Isosceles trapezoids
An isosceles trapezoid has congruent legs. Its base angles are congruent in pairs and its diagonals are congruent. Many bridge supports and handbag shapes are isosceles trapezoids.
Kites
A kite has two pairs of consecutive congruent sides, with no pair of opposite sides congruent. Its diagonals are perpendicular, one diagonal bisects the other, and one pair of opposite angles is congruent. Its area, like a rhombus, is half the product of its diagonals.
Worked example
A trapezoidal garden bed has parallel sides of 8 m and 14 m, 5 m apart. Its area is half of (8 + 14) times 5 = 55 square metres. The midsegment, 11 m, is the average width, so the area also equals 11 x 5, a quick check.
Key takeaways
A rectangle is a parallelogram with right angles and congruent diagonals. A rhombus is a parallelogram with equal sides and perpendicular diagonals that bisect its angles; its area is half the product of the diagonals. A square is both, with all their properties. Every square is a rhombus and a rectangle, but not the reverse. A trapezoid has exactly one pair of parallel bases, a midsegment equal to their average, and area half (b1 + b2) h; isosceles trapezoids have congruent legs and diagonals. Kites have perpendicular diagonals and area half the product of the diagonals.
Flashcard
Parallelogram with four right angles
Check your understanding
Which statement is always true?