Chapter at a glance
- A rectangle is a quadrilateral whose angles are all 90° (equivalently, whose diagonals are equal and bisect each other).
- A square is a rectangle with all sides equal (or a rhombus with one right angle).
- A parallelogram is a quadrilateral with both pairs of opposite sides parallel; opposite sides are equal, opposite angles are equal, adjacent angles sum to 180°, and diagonals bisect each other.
- A rhombus is a quadrilateral with all sides equal; it is a parallelogram whose diagonals bisect each other at right angles and bisect the vertex angles.
- A kite is a quadrilateral with two pairs of adjacent equal sides; one diagonal bisects the other at right angles and bisects the angles at the vertices where the equal sides meet.
- A trapezium is a quadrilateral with at least one pair of parallel sides; base angles on the same side of a non-parallel side are supplementary.
- The sum of the interior angles of any quadrilateral is 360°.
- Relations among quadrilaterals are shown by Venn diagrams: square ⊂ rectangle ⊂ parallelogram; rhombus ⊂ parallelogram; kite and trapezium intersect these sets at specific points.
Definitions, theorems and results
- Rectangle: A quadrilateral in which the angles are all 90° (or whose diagonals are equal and bisect each other).
- Square: A quadrilateral in which all angles are 90° and all sides are equal.
- Parallelogram: A quadrilateral in which opposite sides are parallel.
- Rhombus: A quadrilateral in which all sides have the same length.
- Kite: A quadrilateral ABCD such that AB = BC and CD = DA.
- Trapezium: A quadrilateral with at least one pair of parallel opposite sides.
- Theorem (requires proof): The sum of all angles in any quadrilateral is 360°. (Proof: draw one diagonal to form two triangles, each summing to 180°.)
- All listed properties of rectangles, squares, parallelograms, rhombuses, kites and trapeziums are proved in the chapter via congruence (SAS, AAS, SSS, ASA) and are therefore examinable as proved results.
- Opposite sides of a rectangle/parallelogram/rhombus are parallel and equal.
- Diagonals of a rectangle/square are equal and bisect each other; diagonals of a parallelogram/rhombus bisect each other; diagonals of a rhombus/square are perpendicular and bisect the vertex angles.
- In a parallelogram/rhombus, adjacent angles sum to 180° and opposite angles are equal.
- In an isosceles trapezium, angles adjacent to each equal non-parallel side are equal.
Formula sheet
No numerical formulas appear in the chapter. All results are geometric properties listed above.
Solved-example patterns
- Identify type of quadrilateral or find missing angles/sides given partial measures or diagonal data: draw the figure, mark given equal lengths/angles/parallel lines, apply angle sum 360° or supplementary adjacent angles, or use congruence of triangles formed by a diagonal.
- Construct a rectangle/square/parallelogram/rhombus/kite from diagonal lengths and intersection angle: locate mid-point(s), draw equal diagonals that bisect each other (at 90° if required), join vertices.
- Verify or apply a property (e.g., diagonals bisect angles or are perpendicular): select appropriate triangles, prove congruence using given equal sides/angles or parallel-line alternate angles, deduce corresponding parts.
- Classify relations using Venn diagrams or decide whether a given figure satisfies a definition: check each defining condition in turn.
- Find remaining angles in a trapezium or isosceles trapezium: use co-interior angles on parallel sides sum to 180°, then deduce equal base angles when non-parallel sides are equal.
Common mistakes and exam pitfalls
- Assuming every quadrilateral with equal diagonals that bisect each other is a square (it is only a rectangle; right angle between diagonals is also needed).
- Forgetting that a rhombus (or kite) need not have right angles unless additionally proved.
- Writing incorrect triangle names in congruence statements (e.g., ∆ABD ≅ ∆CBD instead of ∆ABD ≅ ∆CDB) or choosing the wrong congruence criterion.
- Neglecting to prove that all four angles become 90° when diagonals are equal and bisect each other, or assuming opposite angles of a parallelogram are equal without the transversal argument.
- Missing that the angle sum 360° forces the fourth angle of a “three-right-angle” quadrilateral to be 90°.
- Confusing which diagonals bisect angles (only in rhombus/square/kite at specific vertices) or which figures have perpendicular diagonals.
- Overlooking the isosceles-trapezium base-angle property when the non-parallel sides are equal.
- Treating a conjecture obtained by measurement as a proved theorem.