SPI Mathematics · Quadrilaterals

Parallelograms and Other Quadrilaterals

The angles of any quadrilateral add to 360 degrees; in a parallelogram opposite sides and angles are equal and neighbouring angles add to 180 degrees.

All Quadrilaterals notes

Why this matters

Six questions in the bank: two EASY, three MODERATE and one HARD. Each tests one property, such as the angles of a parallelogram, the shape made by joining mid-points, or a trapezium cut by a parallel line, so knowing which shape has which property is most of the work.

Concept 1 of 4: The angle sum of a quadrilateral

A diagonal cuts any quadrilateral into two triangles. Each triangle's angles add to 180 degrees, so the four angles of the quadrilateral add to 360 degrees.

Definition

$\angle A + \angle B + \angle C + \angle D = 360^\circ$ in every quadrilateral.
If the bisectors of $\angle A$ and $\angle B$ meet at P, triangle APB gives $\angle APB = 180^\circ - \frac{\angle A + \angle B}{2}$ .
Since $\angle A + \angle B = 360^\circ - \angle C - \angle D$ , this is the same as $\angle APB = \frac{\angle C + \angle D}{2}$ .
The four bisectors of a quadrilateral make a new quadrilateral whose opposite angles add to $180^\circ$ , because the two halves together use all four angles once.

Angles of a quadrilateral

\angle A + \angle B + \angle C + \angle D = 360^\circ \qquad \angle APB = \frac{\angle C + \angle D}{2}

Worked example

The angles of a quadrilateral are in the ratio 1 : 2 : 3 : 4. Find them.

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The same idea in a practice question:

Practice questionModerate

Example 1 · Quadrilaterals · Parallelograms and Other Quadrilaterals

If bisector of

\angle A

and

\angle B

of a quadrilateral ABCD intersect each other at p,

\angle B

and

\angle C

at Q,

\angle C

and

\angle D

at R and,

\angle D

and

\angle A

at S then PQRS is a

Using 180 for a quadrilateral

180 degrees is the angle sum of a triangle. A quadrilateral is two triangles, so its angles add to

360^\circ

Taking the bisector figure for a rectangle

The bisectors of a general quadrilateral make a figure with opposite angles adding to

180^\circ

. It is a rectangle when the first figure is a parallelogram, but not in general.

Concept 2 of 4: Sides, angles and height of a parallelogram

In a parallelogram each pair of opposite sides is parallel, so each side acts as a transversal of the other pair. That is why neighbouring angles add to 180 degrees and opposite angles come out equal.

Definition

Opposite sides are equal and parallel.
Opposite angles are equal; neighbouring angles add to $180^\circ$ .
The diagonals bisect each other.
The height to a side is the perpendicular from the opposite side. In the right triangle it makes, height $=$ slant side $\times \sin\theta$ , with $\sin 30^\circ = \frac{1}{2}$ , $\sin 45^\circ = \frac{1}{\sqrt{2}}$ , $\sin 60^\circ = \frac{\sqrt{3}}{2}$ .
Area $=$ base $\times$ height.

Parallelogram

\angle A = \angle C, \ \angle B = \angle D, \ \angle A + \angle B = 180^\circ \qquad h = (\text{slant side}) \times \sin\theta \qquad \text{Area} = \text{base} \times h

Worked example

In a parallelogram PQRS,

\angle P = (2x + 30)^\circ

and

\angle Q = (3x - 10)^\circ

. Find all four angles.

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The same idea in a practice question:

Practice questionEasy

Example 2 · Quadrilaterals · Parallelograms and Other Quadrilaterals

In any parallelogram ABCD, the ratio of

\angle

ABC and

\angle

BCD is 1 : 2, the measurement of

\angle

BAD is:

Making neighbouring angles equal

In a parallelogram it is the OPPOSITE angles that are equal. Neighbouring angles add to

180^\circ

, so a ratio 1 : 3 for neighbours gives

45^\circ

and

135^\circ

Using cos for the height

The height is the side opposite the angle in the right triangle, so it is the slant side times

\sin\theta

. The cos gives the piece of the base cut off, not the height.

Drill 1 more on sides, angles and height of a parallelogram

Concept 3 of 4: Special quadrilaterals and the mid-point figure

Each special quadrilateral is a parallelogram with one extra rule, and the diagonals tell them apart. Join the mid-points of the sides of any quadrilateral and you always get a parallelogram; its shape depends on the diagonals of the first figure.

Definition

Mid-point theorem: the line joining the mid-points of two sides of a triangle is parallel to the third side and half of it.
So the mid-point figure of any quadrilateral is a parallelogram with sides parallel to the diagonals.
Perpendicular diagonals give a rectangle; equal diagonals give a rhombus; both give a square.
A square is a special rectangle and a special rhombus, and both of those are special parallelograms.

Shape	What makes it	Diagonals	Joining its mid-points gives
Parallelogram	Both pairs of opposite sides parallel	Bisect each other	A parallelogram
Rectangle	A parallelogram with a right angle	Equal, and bisect each other	A rhombus
Rhombus	A parallelogram with all sides equal	Bisect each other at right angles	A rectangle
Square	A rectangle with all sides equal	Equal, and bisect each other at right angles	A square
Kite	Two pairs of equal neighbouring sides	Cross at right angles	A rectangle
Trapezium	Exactly one pair of parallel sides	Cut each other in the ratio of the parallel sides	A parallelogram A trapezium with equal slant sides has equal diagonals, so its mid-point figure is a rhombus.

The mid-point figure depends only on the diagonals: perpendicular gives right angles, equal gives equal sides.

Practice this conceptself-check · 4 quick reps

The same idea in a practice question:

Practice questionEasy

Example 3 · Quadrilaterals · Parallelograms and Other Quadrilaterals

The figure formed by joining the mid-points of adjacent sides of a Rhombus is a

Thinking the mid-point figure copies the shape

The mid-point figure of a rectangle is a rhombus, not a rectangle. Go by the diagonals: equal diagonals give equal sides, perpendicular diagonals give right angles.

Thinking a square is not a rectangle

A square has four right angles, so it is a rectangle, and four equal sides, so it is a rhombus. When an option says 'rectangle', a square still fits.

Drill 1 more on special quadrilaterals and the mid-point figure

Concept 4 of 4: A line parallel to the sides of a trapezium

Slide a line from the short parallel side to the long one, keeping it parallel. It cuts both slant sides at the same fraction of their length, and its own length grows steadily from the short side to the long side.

Definition

A trapezium has one pair of parallel sides; call them $a$ (shorter) and $b$ (longer).
A line parallel to them that is the fraction $t$ of the way from side $a$ cuts each slant side in the ratio $t : (1 - t)$ , measured from side $a$ .
Its length is $a + (b - a)t$ . Half way, the mid-line is $\frac{a + b}{2}$ .
For a question about two equal perimeters, write each part's perimeter in terms of $t$ and set them equal.

Line parallel to the bases

\text{length} = a + (b - a)\,t \qquad \text{mid-line} = \frac{a + b}{2}

a, bthe shorter and longer parallel sides
tthe fraction of each slant side from side a to the line

Worked example

A trapezium has parallel sides 4 cm and 10 cm. A line parallel to them cuts each slant side in the ratio 1 : 2, measured from the 4 cm side. Find its length.

Practice this conceptself-check · 3 quick reps

The same idea in a practice question:

Practice questionHard

Example 4 · Quadrilaterals · Parallelograms and Other Quadrilaterals

The parallel sides of a trapezoid are 3 cm and 9 cm. The non - parallel sides are 4 cm and 6 cm. A line parallel to the bases divides the trapezoid into two trapezoids of equal perimeters. The ratio in which each of the non - parallel sides is divided is:

Reading the ratio from the wrong end

A ratio is read from the side you start at. A cut 4 : 1 from the short side is 1 : 4 from the long side, so check which end the question measures from.

Taking the mid-line as half the longer side

The mid-line is the average of the two parallel sides,

\frac{a + b}{2}

. For sides 7 cm and 13 cm it is 10 cm, not 6.5 cm.

Summary — formulas & gotchas at a glance

A revision cheat-sheet for the formulas and gotchas above. Click any concept name to jump back to its full explanation.

Formulas (3)

The angle sum of a quadrilateral
Angles of a quadrilateral
$\angle A + \angle B + \angle C + \angle D = 360^\circ \qquad \angle APB = \frac{\angle C + \angle D}{2}$
Sides, angles and height of a parallelogram
Parallelogram
$\angle A = \angle C, \ \angle B = \angle D, \ \angle A + \angle B = 180^\circ \qquad h = (\text{slant side}) \times \sin\theta \qquad \text{Area} = \text{base} \times h$
A line parallel to the sides of a trapezium
Line parallel to the bases
$\text{length} = a + (b - a)\,t \qquad \text{mid-line} = \frac{a + b}{2}$

Reference tables (1)

Special quadrilaterals and the mid-point figure6 rows

Shape	What makes it	Diagonals	Joining its mid-points gives
Parallelogram	Both pairs of opposite sides parallel	Bisect each other	A parallelogram
Rectangle	A parallelogram with a right angle	Equal, and bisect each other	A rhombus
Rhombus	A parallelogram with all sides equal	Bisect each other at right angles	A rectangle
Square	A rectangle with all sides equal	Equal, and bisect each other at right angles	A square
Kite	Two pairs of equal neighbouring sides	Cross at right angles	A rectangle
Trapezium	Exactly one pair of parallel sides	Cut each other in the ratio of the parallel sides	A parallelogram A trapezium with equal slant sides has equal diagonals, so its mid-point figure is a rhombus.

The mid-point figure depends only on the diagonals: perpendicular gives right angles, equal gives equal sides.

Watch out for (8)

Using 180 for a quadrilateral
→ The angle sum of a quadrilateral
Taking the bisector figure for a rectangle
→ The angle sum of a quadrilateral
Making neighbouring angles equal
→ Sides, angles and height of a parallelogram
Using cos for the height
→ Sides, angles and height of a parallelogram
Thinking the mid-point figure copies the shape
→ Special quadrilaterals and the mid-point figure
Thinking a square is not a rectangle
→ Special quadrilaterals and the mid-point figure
Reading the ratio from the wrong end
→ A line parallel to the sides of a trapezium
Taking the mid-line as half the longer side
→ A line parallel to the sides of a trapezium

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