Mini-Lesson

Geometry

4 missions. Guess, learn, then check. Your place is saved.

Mission 1Why can tiles be squares or hexagons but never regular pentagons?
Mission 2Two lookouts see one fire. How do two bearings pinpoint it?
Mission 3A giant ape 10 times a gorilla's height: why would its legs snap?
Mission 4Draw a triangle on a diameter. Why is the top corner always 90°?

The specification points and sources are listed under the lesson: see Sources.

Mission 1

Why can tiles be squares or hexagons but never regular pentagons?

Make a guess: Why can tiles be squares or hexagons but never regular pentagons?

Geometry · angles and polygons

Angle facts and the reasons to write

Name angles with three letters: angle $A BC$ ABC is the angle at $B$ B.
Every angle you find needs a reason written in words. Examiners do not accept "Z angles" or "F angles": use the proper names.

Fact	Reason to write
Angles round a point add to 360°	angles at a point sum to 360°
Angles on a straight line add to 180°	angles on a straight line sum to 180°
Opposite angles at a crossing are equal	vertically opposite angles are equal
Parallel lines: the "Z" pair are equal	alternate angles are equal
Parallel lines: the "F" pair are equal	corresponding angles are equal
Parallel lines: the "C" pair add to 180°	co-interior angles sum to 180°

Worked example Geometry · angles and polygons

Angles on parallel lines, with reasons

Question

In the diagram,

A B

is parallel to

C D

. Angle

EGB = 72°

. Work out angle

x

(angle

G H C

). Give a reason for each step.

1
Use the straight line $EF$ EF: angle $BG H = 180° - 72° = 108°$ BGH = 180° − 72° = 108° (angles on a straight line sum to 180°)
2
Angles $BG H$ BGH and $G H C$ GHC are a "Z" pair between the parallel lines: angle $G H C = 108°$ GHC = 108° (alternate angles are equal)
3
Check another way: angle $A G H = 72°$ AGH = 72° (vertically opposite angle $EGB$ EGB), and $72° + 108° = 180°$ 72° + 108° = 180° for the co-interior pair

Answer

x = 108°

⚠️ Watch out

writing only numbers, or "Z angles", loses the reasoning marks even when the angle is right.

Quick check

?Angles

p

and

q

lie between two parallel lines, on the same side of a line crossing them.

p = 74°

. What is

q

?

Geometry · angles and polygons

Triangles and quadrilaterals

Angles in a triangle sum to 180°. Proof: draw a line through one corner parallel to the opposite side; the two alternate angles and the corner angle make a straight line.
An exterior angle of a triangle equals the sum of the two interior opposite angles.
Isosceles: two equal sides and two equal base angles. Equilateral: three equal sides, every angle 60°.
Angles in a quadrilateral sum to 360°: a diagonal splits it into two triangles.

Quadrilateral	Key properties	Lines of symmetry	Rotational order
Square	4 equal sides, 4 right angles, diagonals equal and cross at 90°	4	4
Rectangle	4 right angles, opposite sides equal, diagonals equal	2	2
Rhombus	4 equal sides, opposite angles equal, diagonals cross at 90°	2	2
Parallelogram	opposite sides parallel and equal, opposite angles equal	0	2
Kite	two pairs of equal adjacent sides, one pair of equal opposite angles, diagonals cross at 90°	1	1
Trapezium	one pair of parallel sides	0 (1 if isosceles)	1

Sort it

?Is each statement always true, or only sometimes true?

Tap a statement, then tap the group it belongs in. Then press Check.

Geometry · angles and polygons

Angles in polygons

From one corner, an $n$ n-sided polygon splits into $n - 2$ n − 2 triangles, so the interior angle sum is

(n - 2) \times 180°

The exterior angles of any polygon sum to 360° (walk round it and you turn once).
In a regular polygon all angles are equal: exterior angle $= 360° \div n$ = 360° ÷ n, and interior angle $= 180° -$ = 180° − exterior angle (they sit on a straight line).
Cambridge: angle sums of irregular polygons are Extended; regular polygons are Core.

Worked example Geometry · angles and polygons

Number of sides from an interior angle

Question

Each interior angle of a regular polygon is 156°. How many sides does it have?

1
Interior and exterior angles sit on a straight line: exterior angle $= 180° - 156° = 24°$ = 180° − 156° = 24°
2
The exterior angles sum to 360°: $n = 360° \div 24° = 15$ n = 360° ÷ 24° = 15
3
Check: $(15 - 2) \times 180° = 2340°$ (15 − 2) × 180° = 2340° and $2340° \div 15 = 156°$ 2340° ÷ 15 = 156°

Answer

15 sides

⚠️ Watch out

dividing 360 by the interior angle (

360 \div 156 \approx 2.31

) gives nonsense. Use the exterior angle.

Calculate

?At one point on a patterned floor, a square tile, a regular hexagon tile and one more tile meet with no gaps and no overlaps. Work out the angle of the third tile at that point.

degrees

Calculate

?The third tile in screen 8 is a regular polygon with interior angle 150°. How many sides does it have?

Mission check

?Why can identical regular pentagons never tile a floor?

Mission 2

Two lookouts see one fire. How do two bearings pinpoint it?

Make a guess: Two lookouts see one fire. How do two bearings pinpoint it?

Geometry · bearings and constructions

Three-figure bearings

A bearing is an angle measured clockwise from north, written with three figures: 065°, 008°, 300°.
"The bearing of $B$ B from $A$ A": stand at $A$ A, face north, turn clockwise until you face $B$ B.
Compass points as bearings: N 000°, NE 045°, E 090°, SE 135°, S 180°, SW 225°, W 270°, NW 315°.
The back bearing (the bearing of $A$ A from $B$ B) differs by exactly 180°, because the north lines at $A$ A and $B$ B are parallel.

Worked example Geometry · bearings and constructions

Finding a back bearing

Question

The bearing of

B

from

A

is 065°. The bearing of

Q

from

P

is 300°. Find the bearing of

A

from

B

, and the bearing of

P

from

Q

.

1
065° is less than 180°, so add 180°: $65° + 180° = 245°$ 65° + 180° = 245°
2
300° is more than 180°, so subtract 180°: $300° - 180° = 120°$ 300° − 180° = 120°
3
Write each answer with three figures

Answer

A

from

B

is 245°;

P

from

Q

is 120°

⚠️ Watch out

"the bearing of

A

from

B

" means you stand at

B

. Reading the question the wrong way round gives the original bearing back.

Calculate

?The bearing of

B

from

A

is 138°. Work out the bearing of

A

from

B

.

degrees

Geometry · bearings and constructions

Scale drawings: fixing a point from two bearings

A scale: "1 cm represents 1 km", or 1 : 100,000.
Two bearings fix a point: draw a north line at each known point, measure each bearing clockwise, draw the lines. The point is where they cross.
Measure the drawn distance; convert with the scale.
Triangle from three sides: draw one side; from its ends draw arcs of the other two lengths; join to the crossing. Leave the arcs.
Extended: know the term perpendicular bisector (at 90° through a segment's midpoint).

Quick check

?On a map with scale 1 : 20,000, two villages are 6.5 cm apart. How far apart are they in real life?

Calculate

?A ship sails from port

P

to buoy

Q

on a bearing of 070°. At

Q

it turns and sails to lighthouse

R

on a bearing of 160°. Work out the size of angle

PQR

.

degrees

Mission check

?Tower

B

is 8 km due east of tower

A

. A fire is on a bearing of 045° from

A

and 315° from

B

. Where is the fire?

Mission 3

A giant ape 10 times a gorilla's height: why would its legs snap?

Make a guess: A giant ape 10 times a gorilla's height: why would its legs snap?

Geometry · congruence and similarity

Congruent or similar?

Congruent shapes are the same shape and size: one fits exactly on the other (after turning or flipping).
Similar shapes are the same shape: corresponding angles are equal and every length is multiplied by the same scale factor $k$ k.
$k = \frac{length on the new shape}{corresponding length on the original}$ k = (length on the new shape)/(corresponding length on the original)
A 4 by 6 rectangle and a 6 by 9 rectangle are similar ( $k = 1.5$ k = 1.5); a 4 by 6 and a 6 by 8 are not.
Triangles with equal angles are similar. Give a reason for each equal angle.

Worked example Geometry · congruence and similarity

A missing length in similar triangles

Question

In triangle ABC, D is on AB and E is on AC with DE parallel to BC. AD = 4 cm, DB = 6 cm and DE = 6 cm. Find BC.

1
Show similarity: angle A is shared; $∠ A D E = ∠ A BC$ ∠ ADE = ∠ ABC (corresponding angles), so the triangles are similar.
2
Use whole sides from A: $A B = 4 + 6 = 10$ AB = 4 + 6 = 10 cm
3
Scale factor from ADE to ABC: $k = \frac{A B}{A D} = \frac{10}{4} = 2.5$ k = AB/AD = 10/4 = 2.5
4
Multiply: $BC = 6 \times 2.5 = 15$ BC = 6 × 2.5 = 15

Answer

BC = 15 cm

⚠️ Watch out

using

\frac{D B}{A D} = 1.5

gives 9 cm. DB is not a side of either triangle.

Calculate

?In triangle PQR, S is on PQ and T is on PR with ST parallel to QR. PS = 5 cm, ST = 4 cm and QR = 10 cm. Find the length SQ.

cm

Geometry · congruence and similarity

Area and volume scale factors (Extended)

If lengths are multiplied by $k$ k, then:
areas (including surface areas) are multiplied by $k^{2}$ k²
volumes (and so masses of the same material) are multiplied by $k^{3}$ k³
Double every length of a cube: area of each face $\times 4$ × 4, volume $\times 8$ × 8.
Given an area or volume factor, work back to $k$ k: area factor 9 means $k = 3$ k = 3; volume factor 8 means $k = 2$ k = 2.

Worked example Geometry · congruence and similarity

Two similar bottles (Extended)

Question

Two bottles are mathematically similar. The small one is 12 cm tall and holds 400 ml. The large one is 18 cm tall. How much does the large one hold?

1
Length scale factor: $k = \frac{18}{12} = 1.5$ k = 18/12 = 1.5
2
Capacity is a volume, so use $k^{3} = 1. 5^{3} = 3.375$ k³ = 1.5³ = 3.375
3
Multiply: $400 \times 3.375 = 1350$ 400 × 3.375 = 1350

Answer

1,350 ml

⚠️ Watch out

400 \times 1.5 = 600

ml uses the length factor and

400 \times 2.25 = 900

ml the area factor.

Calculate

?Two solid cylinders are mathematically similar. Their surface areas are 80 cm² and 180 cm². The smaller one has volume 120 cm³. Find the volume of the larger one.

cm³

Quick check

?A statue is similar to a model and 3 times as tall. Painting the model's surface takes 0.5 litres. How much paint does the statue need?

Mission check

?A giant ape is similar to a gorilla but 10 times as tall. Why would its legs snap?

Mission 4

Draw a triangle on a diameter. Why is the top corner always 90°?

Make a guess: Draw a triangle on a diameter. Why is the top corner always 90°?

Geometry · circle theorems

The parts of a circle

Radius: centre to the circle. Diameter: a chord through the centre, twice the radius.
Chord: a straight line joining two points on the circle. Tangent: a straight line touching the circle at one point.
Arc: part of the circumference. Sector: the slice between two radii and an arc. Segment: the region between a chord and an arc.
Extended: a quadrilateral with all four corners on a circle is a cyclic quadrilateral.

Geometry · circle theorems

Angle in a semicircle (Core and Extended)

AB is a diameter and C is any point on the circle. Then angle $A CB = 9 0^{\circ}$ ACB = 90°: the angle in a semicircle is a right angle.
Why: join C to the centre O. $O A = OB = OC$ OA = OB = OC (radii), so triangles $A OC$ AOC and $BOC$ BOC are isosceles.
Call the base angles $a, a$ a, a and $b, b$ b, b. Triangle $A BC$ ABC gives $2 a + 2 b = 18 0^{\circ}$ 2a + 2b = 180°, so $a + b = 9 0^{\circ}$ a + b = 90°.
Exam reason: "angle in a semicircle is 90°".

Quick check

?AB is a diameter of a circle and C is a point on the circle. Angle

C A B = 3 5^{\circ}

. Find angle

A BC

.

Geometry · circle theorems

Tangents and chords (tangent–radius: Core; the rest: Extended)

A tangent is perpendicular to the radius at the point of contact.
Two tangents from the same outside point are equal in length (by RHS congruence), so they make a kite with the two radii.
The perpendicular from the centre to a chord bisects the chord. The converse holds too: the line from the centre to the midpoint of a chord is perpendicular to it.
Also: a chord's perpendicular bisector passes through the centre; equal chords are equidistant from it.

Calculate

?Exam-style (Extended). PA and PB are tangents from a point P to a circle, centre O, touching it at A and B. Angle

A PB = 5 0^{\circ}

. Find angle

O A B

.

degrees

Geometry · circle theorems

Four angle theorems (Extended)

The angle at the centre is twice the angle at the circumference standing on the same arc.
Angles in the same segment are equal (they stand on the same chord, on the same side).
Opposite angles of a cyclic quadrilateral add up to 180°.
Alternate segment theorem: the angle between a tangent and a chord equals the angle in the alternate segment.
In an exam, quote the theorem in words for every step. "Circle theorem" alone earns no reason mark.

Worked example Geometry · circle theorems

Chasing angles with reasons (Extended)

Question

A, B, C and D lie on a circle, centre O. A is on the major arc BD and C is on the minor arc BD. Angle

BO D = 14 0^{\circ}

. Find angle

B A D

and angle

BC D

, giving reasons.

1
Angle $B A D$ BAD stands on arc BD, like $BO D$ BOD: $B A D = 140 \div 2 = 7 0^{\circ}$ BAD = 140 ÷ 2 = 70° (angle at the centre is twice the angle at the circumference)
2
ABCD is a cyclic quadrilateral: $BC D = 180 - 70 = 11 0^{\circ}$ BCD = 180 − 70 = 110° (opposite angles of a cyclic quadrilateral add up to 180°)

Answer

Angle

B A D = 7 0^{\circ}

and angle

BC D = 11 0^{\circ}

.

⚠️ Watch out

halving 140° for angle

BC D

too. C is on the other side of BD, so it does not stand on the same arc as

BO D

.

Calculate

?Exam-style (Extended). The tangent to a circle at A makes an angle of 64° with the chord AB. C is a point on the circle in the alternate segment, and angle

A BC = 7 1^{\circ}

. Find angle

B A C

.

degrees

Mission check

?AB is a diameter and C is any point on the circle. Why is angle

A CB

always 90°?

Recap

Big ideas

Every angle needs a reason: angles at a point (360°), on a straight line (180°), vertically opposite, alternate, corresponding and co-interior angles.
Triangle 180°, quadrilateral 360°, any polygon $(n - 2) \times 180°$ (n−2) × 180°; exterior angles of a polygon always sum to 360°.
For a regular polygon, find the exterior angle $360° \div n$ 360° ÷ n first, then interior $= 180° -$ = 180° − exterior.
Bearings are measured clockwise from north and written with three figures; a back bearing differs by 180°.
Two bearing lines from two known points cross at exactly one place: that is how you fix a position on a scale drawing.
Similar: equal angles, every length $\times k$ × k. Use whole corresponding sides to find $k$ k.
Lengths $\times k$ × k, areas $\times k^{2}$ × k², volumes $\times k^{3}$ × k³.
Tangent ⟂ radius; two tangents from a point are equal; the perpendicular from the centre bisects a chord.
Angle in a semicircle is 90°; angle at the centre is twice the angle at the circumference; angles in the same segment are equal; opposite angles of a cyclic quadrilateral sum to 180°; alternate segment theorem.

🏆

Mini-lesson complete!

⭐⭐⭐

You've worked through Geometry for Cambridge IGCSE Maths (0580). 🎉

Mission 1Why can tiles be squares or hexagons but never regular pentagons?
Mission 2Two lookouts see one fire. How do two bearings pinpoint it?
Mission 3A giant ape 10 times a gorilla's height: why would its legs snap?
Mission 4Draw a triangle on a diameter. Why is the top corner always 90°?

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Specification points and sources for this lesson are listed at the bottom of the page.