CIE MATH: TOPIC 4 -GEOMETRY

Mathematics · Geometry

Shape & Symmetry

2D Shapes · Polygons · Lines of Symmetry · Rotational Symmetry

📅 15 / 5 / 2026 LO 2.0 — Polygons & Symmetry

Learning Objectives

LO 2.0 — Investigate geometrical properties of polygons
LO 2.0 — Recognise and describe symmetry of 2D shapes

Polygons

Any closed, two-dimensional shape made up of straight lines is called a polygon.

Both triangles and quadrilaterals belong to the family of polygons. If all the sides are the same length and all the interior angles are equal, the shape is called a regular polygon. A square and an equilateral triangle are examples of regular polygons.

Sides	Polygon Name	Sides	Polygon Name
3	Triangle	7	Heptagon
4	Quadrilateral	8	Octagon
5	Pentagon	9	Nonagon
6	Hexagon	10	Decagon

Key Definitions

Line of Symmetry

A line drawn through a shape so that one side is a mirror image of the other side.

Rotational Symmetry

The order is the number of times a shape looks identical to its starting position during one full 360° rotation.

Key Rule: For any regular polygon —
Number of sides = Lines of symmetry = Order of rotational symmetry

Lines of Symmetry in 2D Shapes

Red dashed lines show lines of symmetry. Shapes with 0 lines of symmetry are shown without dashes.

Equilateral Triangle

Lines of symmetry: 3 Rotational order: 3

Square

Lines of symmetry: 4 Rotational order: 4

Rectangle

Lines of symmetry: 2 Rotational order: 2

Rhombus

Lines of symmetry: 2 Rotational order: 2

Kite

Lines of symmetry: 1 Rotational order: 1

Parallelogram

Lines of symmetry: 0 Rotational order: 2

Trapezium

Lines of symmetry: 0 Rotational order: 1

Regular Pentagon

Lines of symmetry: 5 Rotational order: 5

Regular Hexagon

Lines of symmetry: 6 Rotational order: 6

Shape	Lines of Symmetry	Order of Rotational Symmetry
Equilateral Triangle	3	3
Square	4	4
Rectangle	2	2
Rhombus	2	2
Kite	1	1
Parallelogram	0	2
Trapezium	0	1
Regular Pentagon	5	5
Regular Hexagon	6	6

Properties of Quadrilaterals

The table below compares the key properties of the six main quadrilaterals.

Property	Rectangle	Square	Parallelogram	Kite	Rhombus	Trapezium
Opposite sides equal	Yes	Yes	Yes	No	Yes	No
All sides equal length	No	Yes	No	No	Yes	No
All angles equal (90°)	Yes	Yes	No	No	No	No
Diagonals at right angles	No	Yes	No	Yes	Yes	No
Both pairs opp. angles equal	Yes	Yes	Yes	1 pair	Yes	No
Both pairs opp. sides parallel	Yes	Yes	Yes	No	Yes	1 pair
Diagonals equal	Yes	Yes	No	No	No	No

Quick-Reference Facts

Square

Regular polygon. All sides equal, all angles 90°. 4 lines of symmetry, order 4.

Rectangle

Opposite sides equal, all angles 90°. 2 lines of symmetry, order 2.

Rhombus

All sides equal, angles NOT 90°. 2 lines of symmetry (the diagonals), order 2.

Parallelogram

Opposite sides parallel and equal. 0 lines of symmetry, order 2.

Kite

Two pairs of adjacent equal sides. 1 line of symmetry, order 1.

Trapezium

Exactly one pair of parallel sides. Generally 0 lines of symmetry, order 1.

Lesson objectives: Measure and state bearings using three-figure notation (000°–360°) · Calculate bearings and find back bearings · Use a map scale to calculate actual distances

Activity 1 · Starter & Introduction

Compass Directions & Three-Figure Bearings

11:00 – 11:15 · Individual, then class discussion

🧭 Remember

Bearings are always measured clockwise from North and written as three figures e.g. 045°, 270°, 008°.
North = 000° · East = 090° · South = 180° · West = 270°

Part A — Compass directions to bearings

Write the three-figure bearing that matches each compass direction.

Due North

Bearing:

Due East

Bearing:

Due South-West

Bearing:

Due South

Bearing:

Due North-East

Bearing:

Due North-West

Bearing:

Part B — State the bearing of B from A

Use the diagrams below. Measure with a protractor where needed. State the three-figure bearing.

State the bearing of B from A.

Bearing of B from A:

State the bearing of B from A.

Bearing of B from A:

Part C — Convert the angle description to a bearing

A ship sails at 35° east of north. What is its three-figure bearing?

Q10

An aircraft heads at 20° west of south. What is its three-figure bearing?

Activity 2 · Learning in Progress

Back Bearings & Map Scales

11:15 – 11:40 · Pairs, then Think–Pair–Share

Part A — Back Bearings (return bearings)

📐 Rule

Back bearing = forward bearing ± 180°

Bearing A→B = 065°. Find bearing B→A.

Back bearing:

Bearing P→Q = 130°. Find bearing Q→P.

Back bearing:

Bearing X→Y = 305°. Find bearing Y→X.

Back bearing:

Bearing C→D = 250°. Find bearing D→C.

Back bearing:

Part B — Map Scale Activity (Scale 1 : 50 000)

Scale formula

Actual distance = Map distance × 50 000

1 cm on map = 500 m = 0.5 km

Measure map distances (cm) and calculate actual km.

From	To	Map dist (cm)	Actual dist (km)
Landmark A	Landmark B
Landmark B	Landmark C
Landmark A	Landmark C
Your chosen pair

Part C — Think–Pair–Share Multi-Step Problem

A hiker starts at camp C, walks 8 km bearing 072° to A, then 5 km bearing 155° to B.
(a) Bearing from B back to C?
(b) Scale 1 cm : 1 km → draw diagram, find distance BC (nearest 0.5 km).

🤔 Think — My working

👥 Pair — Compare

Bearing B→C:

Distance BC:

Confidence ☆ ☆ ☆ ☆ ☆

Exit Ticket · Plenary

Show What You Know

Name

Date

20 May 2026

Class

Year 10

Q1 — Bearings

Bearing A→B = 115°.
(a) Back bearing B→A?
(b) Plane flies 240 km at 115°, then 180 km at 040°. Describe overall direction.

Back bearing:

Direction description:

Q2 — Map Scale

Map scale 1 : 25 000.
(a) 6 cm on map → actual distance (km)?
(b) Real road 3.5 km → length on map (cm)?

Actual distance:

Map length:

Teacher use:

Q1 /4marks

Q2 /4marks

Total /8marks

Mathematics · Geometry

Shape & Symmetry

2D Shapes · Polygons · Lines of Symmetry · Rotational Symmetry

📅 15 / 5 / 2026 LO 2.0 — Polygons & Symmetry

Learning Objectives

LO 2.0 — Investigate geometrical properties of polygons
LO 2.0 — Recognise and describe symmetry of 2D shapes

Polygons

Any closed, two-dimensional shape made up of straight lines is called a polygon.

Sides	Polygon Name	Sides	Polygon Name
3	Triangle	7	Heptagon
4	Quadrilateral	8	Octagon
5	Pentagon	9	Nonagon
6	Hexagon	10	Decagon

Key Definitions

Line of Symmetry

A line drawn through a shape so that one side is a mirror image of the other side.

Rotational Symmetry

The order is the number of times a shape looks identical to its starting position during one full 360° rotation.

Key Rule: For any regular polygon —
Number of sides = Lines of symmetry = Order of rotational symmetry

Lines of Symmetry in 2D Shapes

Red dashed lines show lines of symmetry. Shapes with 0 lines of symmetry are shown without dashes.

Equilateral Triangle

Lines of symmetry: 3 Rotational order: 3

Square

Lines of symmetry: 4 Rotational order: 4

Rectangle

Lines of symmetry: 2 Rotational order: 2

Rhombus

Lines of symmetry: 2 Rotational order: 2

Kite

Lines of symmetry: 1 Rotational order: 1

Parallelogram

Lines of symmetry: 0 Rotational order: 2

Trapezium

Lines of symmetry: 0 Rotational order: 1

Regular Pentagon

Lines of symmetry: 5 Rotational order: 5

Regular Hexagon

Lines of symmetry: 6 Rotational order: 6

Shape	Lines of Symmetry	Order of Rotational Symmetry
Equilateral Triangle	3	3
Square	4	4
Rectangle	2	2
Rhombus	2	2
Kite	1	1
Parallelogram	0	2
Trapezium	0	1
Regular Pentagon	5	5
Regular Hexagon	6	6

Properties of Quadrilaterals

The table below compares the key properties of the six main quadrilaterals.

Property	Rectangle	Square	Parallelogram	Kite	Rhombus	Trapezium
Opposite sides equal	Yes	Yes	Yes	No	Yes	No
All sides equal length	No	Yes	No	No	Yes	No
All angles equal (90°)	Yes	Yes	No	No	No	No
Diagonals at right angles	No	Yes	No	Yes	Yes	No
Both pairs opp. angles equal	Yes	Yes	Yes	1 pair	Yes	No
Both pairs opp. sides parallel	Yes	Yes	Yes	No	Yes	1 pair
Diagonals equal	Yes	Yes	No	No	No	No

Quick-Reference Facts

Square

Regular polygon. All sides equal, all angles 90°. 4 lines of symmetry, order 4.

Rectangle

Opposite sides equal, all angles 90°. 2 lines of symmetry, order 2.

Rhombus

All sides equal, angles NOT 90°. 2 lines of symmetry (the diagonals), order 2.

Parallelogram

Opposite sides parallel and equal. 0 lines of symmetry, order 2.

Kite

Two pairs of adjacent equal sides. 1 line of symmetry, order 1.

Trapezium

Exactly one pair of parallel sides. Generally 0 lines of symmetry, order 1.

Circle Theorems – Learner Notes

Part I

Circle Theorems I — Angle Properties

Use the following geometrical properties to calculate unknown angles. You are expected to state which property you are using when writing your answer.

Angle in a Semicircle

Any angle inscribed in a semicircle (i.e. the angle subtended by a diameter at the circumference) is always a right angle.

∠ = 90°

Angle Between a Tangent and a Radius

A tangent to a circle meets the radius at the point of contact at exactly 90°. The tangent and radius are perpendicular.

∠ = 90°

Angle at the Centre is Twice the Angle at the Circumference

The central angle is twice any inscribed angle that subtends the same arc. Both angles must be on the same side of the chord.

∠ centre = 2 × ∠ circumference

Angles in the Same Segment are Equal

All inscribed angles subtended by the same arc (in the same segment of the circle) are equal in size.

∠ = ∠ (same segment)

Opposite Angles of a Cyclic Quadrilateral

In a cyclic quadrilateral (all four vertices on the circle), opposite angles are supplementary — they add up to 180°.

∠A + ∠C = 180° | ∠B + ∠D = 180°

Alternate Segment Theorem

The angle between a tangent to a circle and a chord drawn from the point of tangency equals the inscribed angle subtending the same chord on the opposite side (the alternate segment).

∠ tangent–chord = ∠ in alternate segment

You are expected to use these geometrical properties when giving reasons for your answers. Simply writing the angle is not sufficient — name the theorem.

Part II

Circle Theorems II — Symmetry Properties

Apply the symmetry properties of circles to solve problems. As with Part I, you must cite the property used when writing your reasoning.

Equal Chords are Equidistant from the Centre

Two chords of equal length in the same circle are always the same perpendicular distance from the centre of the circle, and vice versa.

chord₁ = chord₂ ⟹ d₁ = d₂ from centre

Perpendicular Bisector of a Chord Passes Through the Centre

The line that bisects any chord at right angles always passes through the centre of the circle. This can be used to locate the centre of a circle.

⊥ bisector of chord → passes through centre

Tangents from an External Point are Equal in Length

If two tangent lines are drawn from a single external point to a circle, the two tangent segments (from the external point to each point of tangency) are equal in length.

PT₁ = PT₂ (from external point P)

You are expected to use these geometrical properties when giving reasons for your answers. Simply writing the angle or length is not sufficient — name the theorem.

⬡ Quick Reference Summary

Angle in a semicircle = 90°
Tangent ⊥ radius = 90°
Central angle = 2 × circumference angle
Angles in same segment are equal
Cyclic quadrilateral: opposite angles sum to 180°
Alternate segment theorem
Equal chords → equidistant from centre
⊥ bisector of chord → through centre
Tangents from external point are equal

Mathematics · Geometry

Shape & Symmetry

2D Shapes · Polygons · Lines of Symmetry · Rotational Symmetry

📅 15 / 5 / 2026 LO 2.0 — Polygons & Symmetry

Learning Objectives

LO 2.0 — Investigate geometrical properties of polygons
LO 2.0 — Recognise and describe symmetry of 2D shapes

Polygons

Any closed, two-dimensional shape made up of straight lines is called a polygon.

Sides	Polygon Name	Sides	Polygon Name
3	Triangle	7	Heptagon
4	Quadrilateral	8	Octagon
5	Pentagon	9	Nonagon
6	Hexagon	10	Decagon

Key Definitions

Line of Symmetry

A line drawn through a shape so that one side is a mirror image of the other side.

Rotational Symmetry

The order is the number of times a shape looks identical to its starting position during one full 360° rotation.

Key Rule: For any regular polygon —
Number of sides = Lines of symmetry = Order of rotational symmetry

Lines of Symmetry in 2D Shapes

Red dashed lines show lines of symmetry. Shapes with 0 lines of symmetry are shown without dashes.

Equilateral Triangle

Lines of symmetry: 3 Rotational order: 3

Square

Lines of symmetry: 4 Rotational order: 4

Rectangle

Lines of symmetry: 2 Rotational order: 2

Rhombus

Lines of symmetry: 2 Rotational order: 2

Kite

Lines of symmetry: 1 Rotational order: 1

Parallelogram

Lines of symmetry: 0 Rotational order: 2

Trapezium

Lines of symmetry: 0 Rotational order: 1

Regular Pentagon

Lines of symmetry: 5 Rotational order: 5

Regular Hexagon

Lines of symmetry: 6 Rotational order: 6

Shape	Lines of Symmetry	Order of Rotational Symmetry
Equilateral Triangle	3	3
Square	4	4
Rectangle	2	2
Rhombus	2	2
Kite	1	1
Parallelogram	0	2
Trapezium	0	1
Regular Pentagon	5	5
Regular Hexagon	6	6

Properties of Quadrilaterals

The table below compares the key properties of the six main quadrilaterals.

Property	Rectangle	Square	Parallelogram	Kite	Rhombus	Trapezium
Opposite sides equal	Yes	Yes	Yes	No	Yes	No
All sides equal length	No	Yes	No	No	Yes	No
All angles equal (90°)	Yes	Yes	No	No	No	No
Diagonals at right angles	No	Yes	No	Yes	Yes	No
Both pairs opp. angles equal	Yes	Yes	Yes	1 pair	Yes	No
Both pairs opp. sides parallel	Yes	Yes	Yes	No	Yes	1 pair
Diagonals equal	Yes	Yes	No	No	No	No

Quick-Reference Facts

Square

Regular polygon. All sides equal, all angles 90°. 4 lines of symmetry, order 4.

Rectangle

Opposite sides equal, all angles 90°. 2 lines of symmetry, order 2.

Rhombus

All sides equal, angles NOT 90°. 2 lines of symmetry (the diagonals), order 2.

Parallelogram

Opposite sides parallel and equal. 0 lines of symmetry, order 2.

Kite

Two pairs of adjacent equal sides. 1 line of symmetry, order 1.

Trapezium

Exactly one pair of parallel sides. Generally 0 lines of symmetry, order 1.

Mathematics · Geometry

Shape & Symmetry

2D Shapes · Polygons · Lines of Symmetry · Rotational Symmetry

📅 15 / 5 / 2026 LO 2.0 — Polygons & Symmetry

Learning Objectives

LO 2.0 — Investigate geometrical properties of polygons
LO 2.0 — Recognise and describe symmetry of 2D shapes

Polygons

Any closed, two-dimensional shape made up of straight lines is called a polygon.

Sides	Polygon Name	Sides	Polygon Name
3	Triangle	7	Heptagon
4	Quadrilateral	8	Octagon
5	Pentagon	9	Nonagon
6	Hexagon	10	Decagon

Key Definitions

Line of Symmetry

A line drawn through a shape so that one side is a mirror image of the other side.

Rotational Symmetry

The order is the number of times a shape looks identical to its starting position during one full 360° rotation.

Key Rule: For any regular polygon —
Number of sides = Lines of symmetry = Order of rotational symmetry

Lines of Symmetry in 2D Shapes

Red dashed lines show lines of symmetry. Shapes with 0 lines of symmetry are shown without dashes.

Equilateral Triangle

Lines of symmetry: 3 Rotational order: 3

Square

Lines of symmetry: 4 Rotational order: 4

Rectangle

Lines of symmetry: 2 Rotational order: 2

Rhombus

Lines of symmetry: 2 Rotational order: 2

Kite

Lines of symmetry: 1 Rotational order: 1

Parallelogram

Lines of symmetry: 0 Rotational order: 2

Trapezium

Lines of symmetry: 0 Rotational order: 1

Regular Pentagon

Lines of symmetry: 5 Rotational order: 5

Regular Hexagon

Lines of symmetry: 6 Rotational order: 6

Shape	Lines of Symmetry	Order of Rotational Symmetry
Equilateral Triangle	3	3
Square	4	4
Rectangle	2	2
Rhombus	2	2
Kite	1	1
Parallelogram	0	2
Trapezium	0	1
Regular Pentagon	5	5
Regular Hexagon	6	6

Properties of Quadrilaterals

The table below compares the key properties of the six main quadrilaterals.

Property	Rectangle	Square	Parallelogram	Kite	Rhombus	Trapezium
Opposite sides equal	Yes	Yes	Yes	No	Yes	No
All sides equal length	No	Yes	No	No	Yes	No
All angles equal (90°)	Yes	Yes	No	No	No	No
Diagonals at right angles	No	Yes	No	Yes	Yes	No
Both pairs opp. angles equal	Yes	Yes	Yes	1 pair	Yes	No
Both pairs opp. sides parallel	Yes	Yes	Yes	No	Yes	1 pair
Diagonals equal	Yes	Yes	No	No	No	No

Quick-Reference Facts

Square

Regular polygon. All sides equal, all angles 90°. 4 lines of symmetry, order 4.

Rectangle

Opposite sides equal, all angles 90°. 2 lines of symmetry, order 2.

Rhombus

All sides equal, angles NOT 90°. 2 lines of symmetry (the diagonals), order 2.

Parallelogram

Opposite sides parallel and equal. 0 lines of symmetry, order 2.

Kite

Two pairs of adjacent equal sides. 1 line of symmetry, order 1.

Trapezium

Exactly one pair of parallel sides. Generally 0 lines of symmetry, order 1.