Big picture
A quadrilateral is a polygon with four sides. Its properties allow us to classify familiar shapes such as rectangles, squares and rhombuses, calculate unknown angles and justify conclusions from limited information. A drawing helps us think, but a shape's appearance is not a proof.
Window frames, floor tiles and flexible four-stick models illustrate different quadrilaterals. A hinged rectangle can lean into a parallelogram while retaining equal opposite sides. That observation explains why equal opposite sides alone do not guarantee right angles. The chapter develops the precise properties that distinguish these shapes.
Learning goals
Distinguish polygons, convex and concave shapes, and regular polygons.
Derive interior-angle and exterior-angle sums.
Calculate angles and numbers of sides of regular polygons.
Classify quadrilaterals using side, angle and diagonal properties.
Explain inclusive relationships among parallelograms, rectangles, rhombuses and squares.
Solve angle and diagonal problems with reasons rather than visual assumptions.
Core ideas
1. Polygons and their boundaries
A polygon is a simple closed figure made of line segments. Simple means that its boundary does not cross itself. A curve is not a polygon side, and an open chain does not enclose a polygon. A triangle has three sides, a quadrilateral four, a pentagon five, a hexagon six and an octagon eight.
A diagonal joins two nonadjacent vertices. From one vertex of an n-sided polygon, exclude that vertex and its two neighbours; there are n − 3 diagonals. Counting from every vertex counts each diagonal twice, so total diagonals = n(n − 3)/2.
Worked example. A hexagon has 6(6 − 3)/2 = 18/2 = 9 diagonals. A quadrilateral has 4(4 − 3)/2 = 2 diagonals. A side is not a diagonal because its endpoints are adjacent.
2. Convex, concave and regular polygons
A convex polygon has each interior angle less than 180°, and all its diagonals lie inside it. A concave polygon has at least one reflex interior angle greater than 180°; at least one diagonal extends outside its interior.
A regular polygon has all sides equal and all angles equal. Both conditions matter. A nonsquare rhombus has equal sides but unequal neighbouring angles, so it is not a regular quadrilateral. A nonsquare rectangle has equal angles but unequal adjacent sides, so it is not regular either. The regular quadrilateral is a square.
In a diagram, label vertices around the boundary in order. For ABCD, sides are AB, BC, CD and DA; diagonals are AC and BD. Joining letters out of order can describe a different or crossed figure.
3. Interior-angle sum
A convex n-sided polygon can be divided from one vertex into n − 2 triangles. Each triangle contributes 180°, giving interior-angle sum = (n − 2) × 180°. This is a sum for the whole polygon, not the size of each angle.
Worked example. A quadrilateral splits into 2 triangles, so its angle sum is 360°. If its angles are 72°, 98°, 115° and x, then x = 360 − (72 + 98 + 115) = 360 − 285 = 75°.
For a pentagon, the total is (5 − 2) × 180° = 540°. A regular pentagon has each angle 540 ÷ 5 = 108°. Do not divide equally unless regularity or equal-angle information is given.
4. Exterior angles and regular polygons
An exterior angle is formed by a side and the extension of an adjacent side. At a convex vertex, the interior and its adjacent exterior angle form a linear pair, so they add to 180°.
Taking one exterior angle at each vertex in the same turning direction gives 360° around a convex polygon. For a regular n-sided polygon, each exterior angle is 360°/n, and each interior angle is 180° − 360°/n.
Worked example. A regular octagon has exterior angle 360 ÷ 8 = 45° and interior angle 180 − 45 = 135°. If each exterior angle of a regular polygon is 24°, then n = 360 ÷ 24 = 15 sides. If each interior angle is 156°, first find exterior = 180 − 156 = 24°, then again obtain 15 sides.
The number of sides must be a whole number at least three. An exterior angle of 50° would give 360/50 = 7.2, so no regular polygon has that exterior angle.
5. A quadrilateral family with clear conventions
Shape | Defining feature | Important consequence |
|---|---|---|
Parallelogram | Both pairs of opposite sides parallel | Opposite sides and angles equal |
Rectangle | A parallelogram with a right angle | All angles are 90° |
Rhombus | A parallelogram with all sides equal | Diagonals perpendicular |
Square | A rectangle with all sides equal | Both rectangle and rhombus properties |
Kite | Two pairs of adjacent equal sides | A diagonal can be an axis of symmetry |
Trapezium | At least one pair of opposite sides parallel | May or may not have a second parallel pair |
This lesson uses the inclusive “at least one pair” convention for trapezium. Some school materials use “exactly one pair”; follow the definition supplied in the question when classifying a parallelogram as a trapezium. Kite definitions can also exclude rhombuses in some books. Calculating from explicitly stated side and diagonal properties avoids ambiguity.
6. Parallelogram properties and reasons
In parallelogram ABCD, AB ∥ CD and BC ∥ AD. A diagonal forms triangles with equal alternate interior angles and a common side; triangle congruence supports equal opposite sides. Parallel-line angle relationships give equal opposite angles and supplementary adjacent angles.
Therefore AB = CD, BC = AD, angle A = angle C, angle B = angle D, and angle A + angle B = 180°. Its diagonals bisect each other: if they meet at O, AO = OC and BO = OD. Bisect does not mean the two whole diagonals are equal.
Worked example. If angle A = 68°, then angle C = 68°, while B = D = 180 − 68 = 112°. If AO = 7 cm and BO = 5 cm, AC = 2 × 7 = 14 cm, BD = 2 × 5 = 10 cm.
7. Rectangle properties
A rectangle is a parallelogram with four right angles. It inherits equal opposite sides and diagonals that bisect one another. Its diagonals are also equal. They are not generally perpendicular unless the rectangle is a square.
Worked example. In rectangle PQRS, diagonal PR = 18 cm and diagonals meet at O. Since QS = PR, QS = 18 cm. Each half-diagonal is 18 ÷ 2 = 9 cm, so PO, RO, QO and SO all equal 9 cm.
Do not conclude that a quadrilateral with equal diagonals must be a rectangle without additional information. An isosceles trapezium can also have equal diagonals. Within the parallelogram family, equal diagonals do identify a rectangle.
8. Rhombus and square
A rhombus has all sides equal, opposite angles equal and adjacent angles supplementary. Its diagonals bisect each other at right angles and each diagonal bisects a pair of opposite angles. Its diagonals need not be equal.
Worked example. A rhombus has perimeter 52 cm. Each side = 52 ÷ 4 = 13 cm. If one interior angle is 70°, the adjacent angle is 180 − 70 = 110°. The diagonal through the 70° vertex bisects it into 35° and 35°.
A square combines rectangle and rhombus properties. All sides are equal, all angles are 90°, and its diagonals are equal perpendicular bisectors of each other. They bisect the vertex angles into 45° each. Every square is a rectangle and a rhombus, but every rectangle or rhombus is not a square.
9. Kite and trapezium properties
In a usual kite ABCD with AB = AD and CB = CD, diagonal AC connects the vertices where equal sides meet. It bisects BD at right angles and bisects angles A and C. Opposite angles B and D are equal. The other diagonal does not generally bisect AC.
Worked example. If the kite's diagonal AC meets BD at O and BD = 16 cm, then BO = OD = 8 cm and angle AOB = 90°. Do not infer AO = OC unless extra information is supplied.
In a trapezium with AB ∥ CD, angles A and D on the same nonparallel side add to 180°, as do B and C. Worked example: A = 76° gives D = 180 − 76 = 104°. An isosceles trapezium has equal nonparallel sides, equal base angles and equal diagonals. These extra properties do not hold for every trapezium.
10. Compare diagonal and symmetry properties
Shape | Diagonals bisect each other? | Diagonals equal? | Diagonals perpendicular? |
|---|---|---|---|
General parallelogram | Yes | Not necessarily | Not necessarily |
Rectangle | Yes | Yes | Not necessarily |
Rhombus | Yes | Not necessarily | Yes |
Square | Yes | Yes | Yes |
Usual nonspecial kite | Only one bisects the other | Not necessarily | Yes |
A nonsquare rectangle has two lines of symmetry through opposite-side midpoints. A nonsquare rhombus has two along its diagonals. A square has four. A general oblique parallelogram has no reflection symmetry, although a half-turn maps it onto itself. A usual kite has one symmetry line along the diagonal joining equal-side vertices; an isosceles trapezium has one through base midpoints.
State “not necessarily” rather than “never” when a special case can possess the property. A square is itself a special rectangle and rhombus.
11. Algebraic angle and classification problems
Worked example. Adjacent parallelogram angles are (3x + 10)° and (2x + 20)°. They are supplementary: 3x + 10 + 2x + 20 = 180. Thus 5x = 150 and x = 30. Angles are 3 × 30 + 10 = 100° and 2 × 30 + 20 = 80°. Opposite angles repeat them.
Worked classification. A parallelogram has perpendicular and equal diagonals. Perpendicular diagonals imply rhombus properties within this family; equal diagonals imply rectangle properties. Together these make it a square. The word parallelogram is important: properties must be used with their stated conditions.
For proof questions, identify the given parallel sides or equality marks, select a valid property and state the conclusion with a reason. Never measure an unscaled drawing to establish an exact angle.
A reliable approach to questions
Draw and label vertices in boundary order.
Record only the stated or validly deduced properties.
Choose angle sum, parallel-line rules or a special-shape property.
Form an equation for unknown angles or lengths.
Check the complete angle sum and any positivity requirements.
For classification, retain family conditions and identify the most specific justified name.
Connections that matter
Triangle congruence and parallel-line angles explain parallelogram properties.
Algebra turns angle relationships into solvable equations.
Mensuration and coordinate graphs use these shape properties for area and length problems.
Common mistakes
Assuming every parallelogram has equal diagonals: this is a rectangle property.
Treating equal sides as enough for a square: right angles are also required.
Confusing bisection with perpendicularity: equal halves and right angles are separate statements.
Trusting a sketch over given conditions: justify every exact conclusion.
Revision checklist
I can classify polygons and describe diagonals.
I can derive angle sums and calculate regular-polygon angles.
I can use quadrilateral side, angle and diagonal properties.
I can explain inclusive relationships among special parallelograms.
I can solve algebraic angle questions with reasons.
I can distinguish a valid converse from an unsupported visual assumption.