Class 8 › Mathematics › Chapter 4
Quadrilaterals
Definitions and properties of rectangles, squares, parallelograms, rhombuses, kites and trapeziums, proved with congruence, plus the angle sum property and constructions using diagonals.
1. What is a Quadrilateral?
Quadrilateral: a closed figure with four straight sides (Latin quadri = four, latus = side). Its angles are the angles between adjacent sides.
- Figures with a curved side are not quadrilaterals.
- A quadrilateral can be concave (one angle bends inward), such as an arrowhead shape; it is still a quadrilateral.
- A diagonal joins opposite vertices. A quadrilateral has two diagonals.
- Congruence tools used throughout: SSS, SAS, ASA, AAS. "Corresponding parts of congruent triangles are equal" gives equal sides and angles.
Conjecture vs proof: measuring a few examples only gives a conjecture (a statement we are confident about but not sure always holds). Only a logical deduction proves a property for every such quadrilateral.
2. Rectangles
Rectangle (first definition): a quadrilateral with all angles $90^\circ$ and opposite sides equal. Below we show it can be defined more simply, or through its diagonals.
Carpenter's problem: two equal strips joined at their midpoints give the corners of a rectangle.
Carpenter's problem: diagonals of a rectangle
Deduction 1 (diagonals are equal): In rectangle $ABCD$, $\triangle BAD$ and $\triangle CDA$ have $AB = CD$, $\angle BAD = \angle CDA = 90^\circ$ and $AD$ common. By SAS they are congruent, so $AC = BD$.
Deduction 2 (diagonals bisect each other): Let diagonals meet at $O$. Since $\angle B = 90^\circ$, $\angle 3 + \angle 1 = 90^\circ$; in $\triangle BCD$, $\angle 3 + \angle 2 = 90^\circ$. So $\angle 1 = \angle 2$. With $\angle AOB = \angle COD$ (vertically opposite) and $AB = CD$, $\triangle AOB \cong \triangle COD$ (AAS). So $OA = OC$ and $OB = OD$.
Deduction 3 (converse): Draw equal diagonals that bisect each other at any angle $x$. The angles between them are $x, x, 180^\circ - x, 180^\circ - x$. In isosceles $\triangle AOB$ base angles are $a = 90^\circ - \tfrac{x}{2}$; in isosceles $\triangle AOD$ base angles are $b = \tfrac{x}{2}$. Each angle of $ABCD$ is $a + b = 90^\circ$. Congruent triangles give $AB = CD$, $AD = BC$, so $ABCD$ is a rectangle.
Second definition: a rectangle is a quadrilateral whose diagonals are equal and bisect each other. (To make a rectangle with an 8 cm diagonal, the other strip must also be 8 cm, and they are joined at their midpoints. Carpenters in Europe and farmers in Mozambique use this method.)
Deduction 4: simplest definition
Let $ABCD$ have all angles $90^\circ$ and join $BD$. As in Deduction 2, $\angle 1 = \angle 2$ (each is $90^\circ - \angle 3$). With $BD$ common and the right angles at $A$ and $C$, $\triangle BAD \cong \triangle DCB$ (AAS). So $AD = CB$ and $DC = BA$.
So a quadrilateral with all four angles $90^\circ$ automatically has equal opposite sides. Hence: a rectangle is a quadrilateral in which all the angles are $90^\circ$. (Write $\triangle BAD \cong \triangle DCB$, not $\triangle CDB$: the vertex order must follow the correspondence.)
Properties of a rectangle
1. All angles are $90^\circ$. 2. Opposite sides are equal. 3. Opposite sides are parallel (since $\angle A + \angle B = 180^\circ$, co-interior angles). 4. Diagonals are equal and bisect each other.
- Finding angles (NCERT exercise): in rectangle $ABCD$ with $\angle CAB = 30^\circ$ (between side $AB$ and diagonal $AC$), we get $\angle CAD = 60^\circ$ (since $\angle A = 90^\circ$) and, because the diagonals are equal and bisect each other (isosceles triangles at $O$), $\angle ABD = 30^\circ$, $\angle ADB = 60^\circ$, $\angle ACD = 30^\circ$, $\angle ACB = 60^\circ$. If the diagonals of a rectangle meet at $110^\circ$, the other angle between them is $70^\circ$; the base angles of the isosceles triangles are $35^\circ$ (opposite the $110^\circ$) and $55^\circ$ (opposite the $70^\circ$).
- Making $90^\circ$ with two equal sticks and thread: cross the sticks at their midpoints, then the thread through the four ends forms a rectangle, so each corner is $90^\circ$.
- Opposite sides parallel and equal is not a definition of a rectangle: a slanted parallelogram has both but no right angles.
3. Squares
Square: a quadrilateral in which all angles are $90^\circ$ and all sides are equal. So every square is a rectangle, but not every rectangle is a square (like "Indian" and "Malayali": one person can be both, as one set sits inside the other).
Deduction 5: diagonals of a square are perpendicular
In square $ABCD$ the diagonals are equal and bisect each other at $O$. In $\triangle BOA$ and $\triangle BOC$: $BA = BC$, $OA = OC$, $BO$ common, so $\triangle BOA \cong \triangle BOC$ (SSS).
Hence $\angle BOA = \angle BOC$, and since they form a straight line, $\angle BOA + \angle BOC = 180^\circ$, so each is $90^\circ$.
Consequence: to build a square, take two equal segments that bisect each other at right angles and join the ends. (Construct a square with a 6 cm diagonal: draw a 6 cm segment, its perpendicular bisector with compass, mark $3$ cm each side of $O$, join. No protractor needed.)
Properties of a square
All the properties of a rectangle, plus: 1. All sides equal. 2. Opposite sides parallel. 3. All angles $90^\circ$. 4. Diagonals are equal and bisect each other at $90^\circ$. 5. Diagonals bisect the angles of the square (each diagonal splits a $90^\circ$ angle into two $45^\circ$ angles, because in $\triangle ADC$, $AD = DC$ and the angle at $D$ is $90^\circ$).
4. Angles in a Quadrilateral
Angle sum property
$$\text{Sum of all angles of any quadrilateral} = 360^\circ$$
Proof
In quadrilateral $SOME$, draw diagonal $SM$ to make $\triangle SEM$ (angles 1, 2, 3) and $\triangle SOM$ (angles 4, 5, 6). Each triangle has angle sum $180^\circ$, so $\angle 1 + \dots + \angle 6 = 360^\circ$.
$(\angle 1 + \angle 4)$, $(\angle 3 + \angle 6)$, $\angle 2$ and $\angle 5$ are the four angles of the quadrilateral, so they add up to $360^\circ$.
- Why not three right angles and a fourth that isn't: the fourth would be $360^\circ - 270^\circ = 90^\circ$ too. So a quadrilateral with three right angles is a rectangle.
- The property also holds for a concave quadrilateral (an arrowhead): join the inner vertex to the opposite one to split it into two triangles.
5. Parallelograms
Parallelogram: a quadrilateral in which both pairs of opposite sides are parallel. Rectangles (and squares) are special parallelograms; but there are slanted ones that are not rectangles.
Construction: draw $AB = 4$ cm and $AD = 5$ cm with $\angle A = 30^\circ$. Draw a line through $D$ parallel to $AB$ and a line through $B$ parallel to $AD$; they meet at $C$.
Deductions 6 to 8
Angles (Deduction 6): $AB \parallel CD$ with transversal $AD$, so $\angle A + \angle D = 180^\circ$ (co-interior angles). Likewise $\angle A + \angle B = 180^\circ$ and $\angle C + \angle D = 180^\circ$. With $\angle A = 30^\circ$: $\angle D = \angle B = 150^\circ$ and $\angle C = 30^\circ$. In general, if one angle is $x$, adjacent ones are $180^\circ - x$ and the opposite one is $x$ again.
Sides (Deduction 7): Join $BD$. In $\triangle ABD$ and $\triangle CDB$: opposite angles of the parallelogram are equal, alternate angles ($AD \parallel BC$, transversal $BD$) are equal, and $BD$ is common. By AAS, $\triangle ABD \cong \triangle CDB$, so $AD = CB$ and $AB = CD$.
Diagonals (Deduction 8): In parallelogram $EASY$ with diagonals meeting at $O$: $AE = YS$ (opposite sides), and alternate angles are equal, so $\triangle AOE \cong \triangle YOS$ (ASA). Hence $OA = OY$ and $OE = OS$.
Properties of a parallelogram
1. Opposite sides are equal. 2. Opposite sides are parallel. 3. Adjacent angles add up to $180^\circ$; opposite angles are equal. 4. Diagonals bisect each other.
- The diagonals of a parallelogram are not necessarily equal, and they do not meet at a fixed angle.
- Converses (all true): a quadrilateral with equal opposite sides is a parallelogram (diagonal $AC$ gives $\triangle ABC \cong \triangle CDA$ by SSS, so alternate angles are equal and opposite sides are parallel); one whose diagonals bisect each other is a parallelogram; one with equal opposite angles is a parallelogram.
- Examples: parallelogram $PEAR$ with $\angle P = 40^\circ$: $\angle E = \angle R = 140^\circ$, $\angle A = 40^\circ$. Parallelogram $PQRS$ with $\angle P = 110^\circ$: $\angle Q = \angle S = 70^\circ$, $\angle R = 110^\circ$.
6. Rhombus (Quadrilaterals with Equal Sides)
Rhombus: a quadrilateral in which all four sides have the same length. Construct it: draw equal sides $AB$ and $AD$ at any angle (not $90^\circ$, else it is a square), then cut arcs of the same radius from $B$ and $D$ to find $C$.
Deductions 9 and 10
Angles (Deduction 9): In rhombus $GAME$, diagonal $AE$ makes four angles $a, b, c, d$. Since $GE = GA$, $a = d$; since $ME = MA$, $b = c$. Also $\triangle GAE \cong \triangle MAE$ (SSS), so $a = b$, $c = d$ and $\angle G = \angle M$. Hence $a = b = c = d$: each diagonal bisects the angles it passes through.
Every rhombus is a parallelogram: alternate angles $a$ are equal, so $EM \parallel GA$ and $GE \parallel AM$.
Example: rhombus with $\angle A = 50^\circ$: in $\triangle ADB$, $a + a + 50^\circ = 180^\circ$, so $a = 65^\circ$. Angles are $50^\circ, 130^\circ, 50^\circ, 130^\circ$ (also $\angle B = \angle D = 180^\circ - 50^\circ = 130^\circ$).
Diagonals (Deduction 10): $\triangle GEO \cong \triangle MEO$ (SSS), so $\angle GOE = \angle MOE$; they form a straight line, so each is $90^\circ$.
Properties of a rhombus
1. All sides equal. 2. Opposite sides parallel. 3. Adjacent angles add to $180^\circ$; opposite angles equal. 4. Diagonals bisect each other. 5. Diagonals bisect the angles. 6. Diagonals meet at $90^\circ$.
- The diagonals of a rhombus are not equal (unless it is a square).
- A quadrilateral with four equal sides and one $90^\circ$ angle is a square (rhombus: adjacent angle is $180^\circ - 90^\circ = 90^\circ$, so all four are $90^\circ$).
- Two congruent equilateral triangles ($8$ cm) joined along a side make a rhombus with angles $60^\circ, 120^\circ, 60^\circ, 120^\circ$ (for sides $4$ cm, all sides are $4$ cm).
7. Kite and Trapezium
Kite: a quadrilateral that can be labelled $ABCD$ with $AB = BC$ and $CD = DA$ (two pairs of equal adjacent sides). Two congruent scalene triangles ($6, 9, 12$ cm) joined along the $12$ cm side as mirror images make a kite with sides $6, 6, 9, 9$.
Kite property
In kite $ABCD$ (with $AB = BC$, $CD = DA$), diagonal $BD$ bisects $\angle ABC$ and $\angle ADC$, and it bisects diagonal $AC$ at right angles ($AO = OC$, $\angle AOB = 90^\circ$).
Why: $\triangle ABD \cong \triangle CBD$ (SSS), so $\angle ABD = \angle CBD$. Then $\triangle AOB \cong \triangle COB$ (SAS: $AB = CB$, equal angles, $BO$ common), so $AO = OC$ and $\angle AOB = \angle COB = 90^\circ$. Only one diagonal bisects the other.
Construct a kite with diagonals $6$ and $8$ cm: draw $PQ = 6$ cm, its perpendicular bisector through $T$, and mark $R$ and $S$ on it $8$ cm apart (with $T$ between them), then join $PR, RQ, QS, SP$.
Trapezium: a quadrilateral with at least one pair of parallel opposite sides. So every parallelogram is also a trapezium.
- Co-interior angles: if $PQ \parallel SR$, then $\angle S + \angle P = 180^\circ$ and $\angle R + \angle Q = 180^\circ$. Example: base angles $135^\circ$ and $105^\circ$ give top angles $45^\circ$ and $75^\circ$.
- Isosceles trapezium: the non-parallel sides are equal. Drop perpendiculars $XY$, $WZ$ from the top corners to the base; $XWZY$ is a rectangle and $\triangle UXY \cong \triangle VWZ$, so $\angle U = \angle V$. Angles opposite to the equal sides are equal (each pair of base angles is equal). Example: top-left angle $100^\circ$ gives top-right $100^\circ$ and both base angles $80^\circ$.
- Isosceles trapeziums are not parallelograms: the other pair of opposite sides need not be parallel.
8. The Quadrilateral Family
Rhombus = kite ∩ parallelogram. Square = rhombus ∩ rectangle. Every parallelogram is a trapezium.
- Square $\subset$ rectangle $\subset$ parallelogram $\subset$ trapezium. Also square $\subset$ rhombus $\subset$ parallelogram, and rhombus $\subset$ kite.
- A quadrilateral that is both a kite and a parallelogram is a rhombus (this includes the square).
- Every rhombus is a kite, but not every kite is a rhombus (a kite has only two pairs of adjacent equal sides).
- A shape that is both a kite and a rectangle: only the square.
| Quadrilateral | Diagonals equal? | Diagonals bisect each other? | Diagonals perpendicular? | Diagonals bisect the angles? |
|---|---|---|---|---|
| Parallelogram | Not always | Yes | Not always | No |
| Rectangle | Yes | Yes | Not always | No |
| Rhombus | Not always | Yes | Yes | Yes |
| Square | Yes | Yes | Yes | Yes |
| Kite | Not always | Only one bisects the other | Yes | The main diagonal bisects its two angles |
9. Constructions and Activities
| To build a | Draw two diagonals that… | Example |
|---|---|---|
| Parallelogram | bisect each other (any angle, any lengths) | $7$ cm and $5$ cm at $140^\circ$: draw $AB = 7$ cm, midpoint $O$, a line at $140^\circ$, mark $2.5$ cm each side of $O$ |
| Rectangle | are equal and bisect each other | both $8$ cm, at $30^\circ$, $40^\circ$, $90^\circ$ or $140^\circ$ (any angle) |
| Rhombus | bisect each other at $90^\circ$ | $4$ cm and $5$ cm: $AB = 5$ cm, perpendicular through $O$, mark $2$ cm each side |
| Square | are equal and bisect each other at $90^\circ$ | diagonal $6$ cm: $OC = OD = 3$ cm |
| Kite | are perpendicular, one bisecting the other | $6$ cm and $8$ cm |
- Circle fact: two perpendicular diameters $PL$ and $AM$ of a circle (centre $O$) are equal and bisect each other at $90^\circ$, so $APML$ is a square.
- Midpoints of a square's sides: if $CASE$ has side $x$, the midpoint quadrilateral $UVWX$ has all sides $\tfrac{x}{\sqrt{2}}$ (Pythagoras on right triangles with legs $\tfrac{x}{2}$) and all angles $90^\circ$ (the $45^\circ$ angles at the corners leave $90^\circ$). So it is a square. Other inner squares with vertices on the sides of the outer square can be tilted at any angle (figure b).
- Geoboard: two equal rubber bands crossing perpendicularly at their midpoints give a square. Extending one by $2$ cm on both sides gives a rhombus (diagonals still bisect each other at $90^\circ$ but are no longer equal).
- Joining two congruent triangles:
• Two equilateral triangles: a rhombus.
• Two isosceles triangles ($8, 8, 6$): a rhombus (join along the $6$ cm side, mirror image) or a parallelogram (join along an $8$ cm side, rotated copy).
• Two scalene triangles ($6, 9, 12$): a kite (mirror image along a side) or a parallelogram (rotated copy along a side). - Paper folding ("Which Quad?"): folding a sheet into quarters and creasing the middle corner gives a rhombus of creases.
- Rectangles PAIR and RODS: in right triangle $RIO$, $\angle ROI = 90^\circ - 30^\circ = 60^\circ$; since $\angle ROD = 90^\circ$, $\angle IOD = 90^\circ - 60^\circ = 30^\circ$.
10. Summary and Quick Revision
| Shape | Definition | Key properties |
|---|---|---|
| Rectangle | All angles $90^\circ$ | Opposite sides equal and parallel; diagonals equal and bisect each other |
| Square | All angles $90^\circ$ and all sides equal | Opposite sides parallel; diagonals equal, bisect each other at $90^\circ$ and bisect the angles |
| Parallelogram | Opposite sides parallel | Opposite sides equal; adjacent angles sum $180^\circ$, opposite angles equal; diagonals bisect each other |
| Rhombus | All sides equal | A parallelogram; diagonals bisect each other at $90^\circ$ and bisect the angles |
| Kite | Two pairs of equal adjacent sides | One diagonal is the perpendicular bisector of the other and bisects the angles at its ends |
| Trapezium | At least one pair of parallel sides | Co-interior angles along each leg add to $180^\circ$; isosceles: base angles equal |
| Any quadrilateral | Angle sum $= 360^\circ$ |
True or false (NCERT Figure it Out):
• Equal diagonals that bisect each other must give a square: False (a rectangle also has them).
• Three right angles means a rectangle: True (the fourth is $360^\circ - 270^\circ = 90^\circ$).
• Diagonals bisect each other means a parallelogram: True.
• Perpendicular diagonals means a rhombus: False (a kite also has them).
• Equal opposite angles means a parallelogram: True.
• All angles equal means a rectangle: True (each is $90^\circ$).
• Isosceles trapeziums are parallelograms: False (the legs need not be parallel).
• Equal diagonals that bisect each other must give a square: False (a rectangle also has them).
• Three right angles means a rectangle: True (the fourth is $360^\circ - 270^\circ = 90^\circ$).
• Diagonals bisect each other means a parallelogram: True.
• Perpendicular diagonals means a rhombus: False (a kite also has them).
• Equal opposite angles means a parallelogram: True.
• All angles equal means a rectangle: True (each is $90^\circ$).
• Isosceles trapeziums are parallelograms: False (the legs need not be parallel).
Common mistakes:
• Saying a rhombus has equal diagonals (only a square does).
• Saying a parallelogram's diagonals are equal or perpendicular.
• Assuming a rectangle's diagonals are perpendicular (only a square's are).
• Thinking a kite's two diagonals bisect each other (only one does).
• Forgetting that every square is also a rectangle, a rhombus, a parallelogram, a trapezium and a kite.
• Writing congruence with the wrong vertex order ($\triangle BAD \cong \triangle CDB$ is wrong; use $\triangle CDA$).
• Using "equal opposite sides and parallel" as a definition of a rectangle.
• Concluding from a few measured cases (a conjecture) that a property is proved.
• Saying a rhombus has equal diagonals (only a square does).
• Saying a parallelogram's diagonals are equal or perpendicular.
• Assuming a rectangle's diagonals are perpendicular (only a square's are).
• Thinking a kite's two diagonals bisect each other (only one does).
• Forgetting that every square is also a rectangle, a rhombus, a parallelogram, a trapezium and a kite.
• Writing congruence with the wrong vertex order ($\triangle BAD \cong \triangle CDB$ is wrong; use $\triangle CDA$).
• Using "equal opposite sides and parallel" as a definition of a rectangle.
• Concluding from a few measured cases (a conjecture) that a property is proved.