Properties of quadrilaterals

Learning outcome

Classify quadrilaterals and apply side, angle and diagonal properties without assuming every quadrilateral is cyclic.

Concepts and assumptions

Use convex quadrilaterals with four distinct vertices and no 180° interior angle. In ABCD, letters name consecutive vertices; AC and BD are diagonals. A diagonal divides the figure into two triangles, so its interior angles total 360°.

Parallelogram: Both pairs of opposite sides are parallel and equal. Opposite angles are equal, adjacent angles total 180°, and diagonals bisect each other.

Rectangle: A parallelogram with four right angles. Diagonals are equal and bisect each other, but need not be perpendicular.

Rhombus: A parallelogram with four equal sides. Diagonals bisect each other perpendicularly and bisect the vertex angles, but need not be equal.

Square: Both a rectangle and a rhombus; it has all their properties.

Trapezium: Here, exactly one pair of opposite sides is parallel. Angles along either nonparallel side total 180°; the diagonals need not bisect each other.

Kite: Two pairs of adjacent sides are equal. If AB = AD and CB = CD, AC perpendicularly bisects BD; BD need not bisect AC.

A convex quadrilateral is cyclic when all four vertices lie on one circle. Its opposite angles total 180°; conversely, this supplementary-angle condition guarantees cyclicity. Do not confuse this with a parallelogram’s adjacent-angle rule.

Worked examples

Example 1 — Adjacent parallelogram angles. In parallelogram ABCD, ∠A = (3x + 10)° and ∠B = (2x − 5)°. Adjacent angles are supplementary: 3x + 10 + 2x − 5 = 180. Therefore 5x = 175 and x = 35. Thus ∠A = ∠C = 115° and ∠B = ∠D = 65°.

Example 2 — Rhombus side from diagonals. In rhombus ABCD, AC = 16 cm and BD = 30 cm intersect at O. Perpendicular bisection gives AO = 8 cm and BO = 15 cm. Right triangle AOB gives AB² = 8² + 15² = 289, so every side is 17 cm.

Example 3 — Opposite cyclic angles. All vertices of convex ABCD lie on one circle. Given ∠A = (3y + 5)° and ∠C = (2y + 25)°, opposite angles give 5y + 30 = 180. Hence y = 30, ∠A = 95° and ∠C = 85°. Individual values of ∠B and ∠D remain undetermined.

Common mistakes

Do not transfer square properties to every rectangle or rhombus. Equal diagonals alone do not prove a rectangle. Opposite angles are not supplementary in every quadrilateral. A sketch’s appearance proves neither parallelism nor equal lengths.

Practice questions

1. Convex ABCD has ∠A = 82°, ∠B = 104° and ∠C = 96°. Find ∠D.
2. Rectangle ABCD has AC = 26 cm; its diagonals meet at O. Find BD and BO.
3. Trapezium ABCD has AB ∥ CD, ∠A = 68° and ∠C = 117°. Find ∠B and ∠D.
4. A parallelogram is also cyclic. Find its angles. Must it be a square?

Worked solutions

1. The angle sum is 360°. Therefore ∠D = 360° − (82° + 104° + 96°) = 360° − 282° = 78°.
2. Rectangle diagonals are equal, so BD = AC = 26 cm. They bisect each other, giving BO = 26 ÷ 2 = 13 cm.
3. Parallel bases give ∠B + ∠C = 180° and ∠A + ∠D = 180°. Thus ∠B = 63° and ∠D = 112°.
4. Opposite angles are equal by the parallelogram property and supplementary by cyclicity. Twice either angle is 180°, so all angles are 90°. It is a rectangle; equal sides are not guaranteed, so it need not be a square.