A frame has two equal diagonals. Can you tighten its fasteners now? An isosceles trapezoid also has equal diagonals. The answer depends on what else you have established about the shape.

A rectangle is a parallelogram with a right angle. A square is both a rectangle and a rhombus. Equal diagonals become a test for a rectangle only with an additional condition, such as the diagonals bisecting each other. We use parallel lines and triangle congruence.

A family that inherits properties

All quadrilaterals here are simple, convex and nondegenerate; A,B,C,D follow the boundary in order. Diagonals AC and BD join opposite vertices. A diagonal divides the figure into two triangles, so its interior angles sum to 360°.

  • Parallelogram: both pairs of opposite sides are parallel.
  • Rectangle: all angles are right angles; it is a special parallelogram.
  • Rhombus: all sides are equal; it too is a parallelogram.
  • Square: all sides are equal and all angles are right angles; it belongs to both preceding classes.
  • Trapezoid: in this course, exactly one pair of opposite sides is parallel. Those sides are its bases; the others are its legs.

Under this convention, a parallelogram is not a trapezoid. Other sources use “at least one pair”: their hierarchy differs, though the geometry does not. We use the American name trapezoid; British trapezium can mean the same figure. AMSI on rhombuses and trapezia.

Convex quadrilateralsParallelogramsTrapezoidsRectanglesRhombusesSquares

Rectangular frame

ABCD8060

Equal diagonals: not yet a rectangle

ABCDAC = BD

Perpendicular: not yet a rhombus

ABCDOAO ≠ OC
Arrows go from a general class to a special type. Squares are both rectangles and rhombuses. Trapezoids have exactly one parallel pair. Separate panels show a frame with sides 80 and 60, a trapezoid with equal diagonals but no right angles, and a kite with perpendicular diagonals that do not bisect each other. Uniform scale within each figure; scales differ between panels.

Why parallelograms are useful

Their opposite sides and angles are equal, adjacent angles sum to 180°, and the diagonals bisect each other at their intersection O:

AO = OC, BO = OD.

One diagonal makes two congruent triangles: angles formed with the parallel sides are equal, and the diagonal is shared. This gives equal opposite sides. Congruent triangles around O then give equal diagonal halves. These follow from parallelism, not from a neatly drawn picture.

To recognise a parallelogram among convex quadrilaterals, any one of these conditions suffices:

  1. Both pairs of opposite sides are equal.
  2. The same pair of opposite sides is both equal and parallel.
  3. The diagonals bisect each other.
  4. Both pairs of opposite angles are equal.

In condition two, you cannot use equality of one pair and parallelism of the other. AMSI tests and proofs.

What each special shape adds

  • Rectangle. Equal and bisect each other. For a known parallelogram, it suffices to check: One right angle or equal diagonals.

  • Rhombus. Perpendicular and bisect each other; each bisects the angles at its endpoints. For a known parallelogram, it suffices to check: Two adjacent sides equal or perpendicular diagonals.

  • Square. Equal, perpendicular and bisect each other. For a known parallelogram, it suffices to check: Both the rectangle and rhombus checks.

Why do equal diagonals work in a parallelogram? Triangles ABC and BAD share AB, with BC=AD and AC=BD. By SSS, angles ABC and BAD are equal. They are also adjacent angles of a parallelogram and sum to 180°, so each is 90°.

Why does perpendicularity give a rhombus? If the diagonals of a parallelogram are perpendicular, right triangles AOB and AOD have equal legs: AO is shared and BO=OD. Hence AB=AD; equality of opposite sides gives all four equal sides.

A frame: a sufficient check and its measurement limits

Suppose both pairs of opposite sides of a flat four-bar frame have lengths 80 cm and 60 cm respectively. That already makes it a parallelogram. Equal diagonals now prove it is rectangular. Alternatively, check one right angle. We do not need a numerical diagonal length here.

If the side lengths have not been established, an alternative is to check that both diagonals bisect each other and are equal. Equal diagonals alone do not suffice: the diagram shows an isosceles trapezoid as a counterexample. Nor do perpendicular diagonals alone prove a rhombus: in the illustrated kite, they do not bisect each other.

Physical measurements have errors. The theorem provides an exact criterion for an ideal flat shape; acceptable deviations in a real product must be specified separately.

The trapezoid is a separate branch

With bases AB∥CD, the angles along each leg sum to 180°: ∠A+∠D=180° and ∠B+∠C=180°. An isosceles trapezoid has equal legs; its angles at either base are equal, and its diagonals are equal. The converse tests also work for a trapezoid with exactly one parallel pair: equal diagonals or equal angles at one base imply equal legs.

A right trapezoid has one leg perpendicular to the bases, giving two right angles. Conditions for circles around or inside quadrilaterals are covered in G008; we need not introduce them again here.