Sec 2 · G1 (N(T))

Quadrilaterals and polygons

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1. Quadrilaterals

Quadrilateral	Sides	Angles	Diagonals
parallelogram	opposite sides parallel and equal	opposite angles equal	bisect each other
rectangle	opposite sides parallel and equal	all $90^\circ$	equal, bisect each other
rhombus	all equal	opposite angles equal	bisect each other at $90^\circ$
square	all equal	all $90^\circ$	equal, bisect each other at $90^\circ$
trapezium	one pair parallel
isosceles trapezium	one pair parallel; the other two sides equal	base angles equal	equal
kite	two pairs of equal adjacent sides	one pair equal	cross at $90^\circ$ ; one bisects the other

📖 Worked example

$ABCD$ is a parallelogram. Find $x$ .

Step 1 of 2

Which angle fact gives x?

x^\circ + 113^\circ = 180^\circ

(int. ∠s,

AB \parallel DC

)

Angles in a ratio	Share	Example
$\angle A : \angle B : \angle C : \angle D$	$360^\circ$	$3 : 4 : 5 : 6$ : $18$ units $\rightarrow 360^\circ$ , $1$ unit $\rightarrow 20^\circ$
parallelogram, $\angle A : \angle B$	$180^\circ$ (int. $\angle$ s)	$2 : 3$ : $1$ unit $\rightarrow 36^\circ$

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2. Special quadrilaterals

The seven special quadrilaterals with their marks and dashed diagonals: a square (four equal sides, four right angles, diagonals at right angles); a rectangle (opposite sides equal, four right angles); a rhombus (four equal sides, diagonals at right angles); a parallelogram (both pairs of opposite sides parallel); a kite (two pairs of equal sides next to each other, diagonals at right angles); a trapezium (one pair of parallel sides); an isosceles trapezium (one pair of parallel sides, the other two sides equal, base angles both 64°).

Property	square	rectangle	rhombus	parallelogram	kite	trapezium	isosceles trapezium
all sides equal	✓	✗	✓	✗	✗	✗	✗
both pairs of opposite sides parallel	✓	✓	✓	✓	✗	✗	✗
only one pair of parallel sides	✗	✗	✗	✗	✗	✓	✓
all angles $90^\circ$	✓	✓	✗	✗	✗	✗	✗
opposite angles equal	✓	✓	✓	✓	✗	✗	✗
diagonals bisect each other	✓	✓	✓	✓	✗	✗	✗
diagonals equal	✓	✓	✗	✗	✗	✗	✓
diagonals cross at $90^\circ$	✓	✗	✓	✗	✓	✗	✗
diagonals bisect the angles	✓	✗	✓	✗	✗	✗	✗

📖 Worked example

The marks on the diagram show what is known about quadrilateral $ABCD$ . Write down the most special name of $ABCD$ .

Step 1 of 4

Equal sides (the ticks)

Equal sides: all four.

📖 Worked example

$ABCD$ is a rhombus. Its diagonals meet at $O$ . $AC = 12$ cm and $BD = 16$ cm. Find $AB$ .

Step 1 of 2

The diagonals of a rhombus bisect each other at right angles

OA = 6

cm,

OB = 8

\angle AOB = 90^\circ

📖 Worked example

$ABCD$ is a rectangle. Its diagonals meet at $O$ , and $\angle OAB = 28^\circ$ . Find $x$ .

Step 1 of 2

Find ∠OBA and give the reason

\angle OBA = 28^\circ

(base ∠s of isos. △)

The angles show	Sides not all equal	Four equal sides
all four angles $90^\circ$	rectangle	square
both pairs of opposite angles equal	parallelogram	rhombus
only $\angle A + \angle D = 180^\circ$ ( $AB \parallel DC$ )	trapezium

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1. Quadrilaterals

Quadrilateral	Sides	Angles	Diagonals
parallelogram	opposite sides parallel and equal	opposite angles equal	bisect each other
rectangle	opposite sides parallel and equal	all $90^\circ$	equal, bisect each other
rhombus	all equal	opposite angles equal	bisect each other at $90^\circ$
square	all equal	all $90^\circ$	equal, bisect each other at $90^\circ$
trapezium	one pair parallel
isosceles trapezium	one pair parallel; the other two sides equal	base angles equal	equal
kite	two pairs of equal adjacent sides	one pair equal	cross at $90^\circ$ ; one bisects the other

📖 Worked example

$ABCD$ is a parallelogram. Find $x$ .

Step 1 of 2

Which angle fact gives x?

x^\circ + 113^\circ = 180^\circ

(int. ∠s,

AB \parallel DC

)

Angles in a ratio	Share	Example
$\angle A : \angle B : \angle C : \angle D$	$360^\circ$	$3 : 4 : 5 : 6$ : $18$ units $\rightarrow 360^\circ$ , $1$ unit $\rightarrow 20^\circ$
parallelogram, $\angle A : \angle B$	$180^\circ$ (int. $\angle$ s)	$2 : 3$ : $1$ unit $\rightarrow 36^\circ$

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2. Special quadrilaterals

Property	square	rectangle	rhombus	parallelogram	kite	trapezium	isosceles trapezium
all sides equal	✓	✗	✓	✗	✗	✗	✗
both pairs of opposite sides parallel	✓	✓	✓	✓	✗	✗	✗
only one pair of parallel sides	✗	✗	✗	✗	✗	✓	✓
all angles $90^\circ$	✓	✓	✗	✗	✗	✗	✗
opposite angles equal	✓	✓	✓	✓	✗	✗	✗
diagonals bisect each other	✓	✓	✓	✓	✗	✗	✗
diagonals equal	✓	✓	✗	✗	✗	✗	✓
diagonals cross at $90^\circ$	✓	✗	✓	✗	✓	✗	✗
diagonals bisect the angles	✓	✗	✓	✗	✗	✗	✗

📖 Worked example

The marks on the diagram show what is known about quadrilateral $ABCD$ . Write down the most special name of $ABCD$ .

Step 1 of 4

Equal sides (the ticks)

Equal sides: all four.

📖 Worked example

$ABCD$ is a rhombus. Its diagonals meet at $O$ . $AC = 12$ cm and $BD = 16$ cm. Find $AB$ .

Step 1 of 2

The diagonals of a rhombus bisect each other at right angles

OA = 6

cm,

OB = 8

\angle AOB = 90^\circ

📖 Worked example

$ABCD$ is a rectangle. Its diagonals meet at $O$ , and $\angle OAB = 28^\circ$ . Find $x$ .

Step 1 of 2

Find ∠OBA and give the reason

\angle OBA = 28^\circ

(base ∠s of isos. △)

The angles show	Sides not all equal	Four equal sides
all four angles $90^\circ$	rectangle	square
both pairs of opposite angles equal	parallelogram	rhombus
only $\angle A + \angle D = 180^\circ$ ( $AB \parallel DC$ )	trapezium

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