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Cambridge IGCSE Maths (0580) · Geometry
Mini-Lesson

Geometry

This mini-lesson covers Geometry for Cambridge IGCSE Mathematics (0580): angle properties, polygons, circle theorems, congruence & similarity, symmetry and constructions (Extended content included).

Honest heads-up: these games test recall of angle facts and theorems, not full geometric reasoning. Give full reasons in the exam.

Answer as you go and collect ⭐ stars. Press Start.

Angles

Angle properties

Key facts: angles on a straight line sum to 180°, angles at a point sum to 360°, and the angles in a triangle sum to 180°. With parallel lines, corresponding and alternate angles are equal.

triangle angles = 180°quadrilateral angles = 360°

Isosceles: two equal sides give two equal base angles.

Calculate

Your turn

1Two angles of a triangle are 55° and 68°. Work out the third angle in degrees.
°
Hint: angles sum to 180°, so 180 − 55 − 68.
Polygons

Polygons

The interior angles of a polygon with n sides sum to (n − 2) × 180°. The exterior angles always sum to 360°.

interior sum = (n − 2) × 180°each exterior angle of a regular n-gon = 360° ÷ n
Worked example

Pentagon: (5 − 2) × 180 = 540°. Regular pentagon exterior angle = 360 ÷ 5 = 72°.

Calculate

Your turn

2Work out the sum of the interior angles of a hexagon (6 sides).
°
Hint: (6 − 2) × 180.
Calculate

Your turn

3A regular polygon has an exterior angle of 30°. How many sides does it have?
Hint: number of sides = 360 ÷ exterior angle = 360 ÷ 30.
Circle theorems

Circle theorems

Core theorems: the angle at the centre is twice the angle at the circumference on the same arc; the angle in a semicircle is 90°; angles in the same segment are equal; opposite angles of a cyclic quadrilateral sum to 180°.

angle at centre = 2 × angle at circumferenceangle in a semicircle = 90°

Tangent facts: a tangent meets a radius at 90°, and two tangents from a point are equal in length.

Calculate

Your turn

4An arc subtends an angle of 40° at the circumference. What is the angle it subtends at the centre?
°
Hint: angle at centre = 2 × angle at circumference.
Quick check

Quick check

?In a cyclic quadrilateral, one angle is 105°. What is the size of the opposite angle?
Congruence & similarity

Congruence & similarity

Congruent shapes are identical (same size and shape). Similar shapes have the same shape but a scale factor difference — corresponding angles equal, sides in proportion.

scale factor k: lengths ×k, areas ×k², volumes ×k³congruence tests: SSS, SAS, ASA, RHS
Worked example

Two similar triangles with scale factor 3: a 5 cm side maps to 15 cm. Areas scale by 3² = 9.

Calculate

Your turn

5Two similar rectangles have a length scale factor of 4. The small one has area 6 cm². Work out the area of the large one.
cm²
Hint: area scale factor = 4² = 16, so 6 × 16.
Quick check

Quick check

?Two triangles are similar with a length scale factor of 2. How do their areas compare?
Match game

Polygon ⇄ angle sum

Tap a polygon, then its interior-angle sum.

Symmetry & constructions

Symmetry & constructions

A shape has line symmetry (mirror lines) and rotational symmetry (order = how many times it looks the same in a full turn). Constructions use compass and straight edge — e.g. a perpendicular bisector or an angle bisector.

Worked example

A square has 4 lines of symmetry and rotational symmetry of order 4.

Regular n-gon: n lines of symmetry and rotational symmetry of order n.

Calculate

Your turn

6A regular hexagon has how many lines of symmetry?
Hint: a regular n-gon has n lines of symmetry.
Sort it

Right angle or not?

Sort each angle: is it exactly 90° (a right angle) or not? Tap it, then its box.

⊾ Right angle (90°)

✗ Not 90°

Recap

Key points

Angles: line 180°, point 360°, triangle 180°; parallel-line rules.

Polygons: interior sum (n − 2)×180°; exterior sum 360°.

Circle theorems: centre = 2 × circumference; semicircle 90°; cyclic quad opposite angles 180°.

Similarity: lengths ×k, areas ×k², volumes ×k³.

Symmetry: lines & rotational order; regular n-gon → n of each.

You've covered the Geometry essentials for Cambridge IGCSE Maths (0580). Press Finish to see your score.

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