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GEOMETRY LAB
Learn it by playing it
Four short missions on two live stages. Steer the angle sliders, watch the triangle and the parallel lines react, and figure out the rules. When a mission glows green, read why it worked.
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Mission 1 of 4
The 180 game
Two angle sliders set corners A and B of a live triangle; angle C is whatever is left. Steer the sliders until angle C is exactly 45°.
A 60° · B 70° · C 50° · exterior at C 130°
Before you play: in a triangle, A = 50° and B = 90°. What is angle C?
Currently C = 50° - too big. C shrinks when A or B grows.
Study notes
Geometrical vocabulary C4.1 / E4.1
Angles are classified by size: acute (under 90°), right (exactly 90°), obtuse (90° to 180°), reflex (180° to 360°).
Triangles are classified by sides and angles: equilateral (all sides equal), isosceles (two sides equal), scalene (no sides equal), right-angled (one 90° angle).
Know the special quadrilaterals - square, rectangle, parallelogram, rhombus, trapezium, kite - and which sides are parallel or equal in each.
Polygon names by side count: pentagon (5), hexagon (6), octagon (8), decagon (10); a regular polygon has all sides and all angles equal.
Circle vocabulary: centre, radius, diameter, circumference, semicircle, chord, tangent, arc, sector, segment. A chord joins two points on the circumference; a tangent touches the circle at exactly one point.
Similar shapes have the same angles and proportional sides; congruent shapes are identical in shape and size. The scale factor maps one similar shape onto another.
Constructions and nets C4.2 / E4.2
Use a ruler for every straight edge; measure and draw angles with a protractor.
To construct a triangle from three side lengths, draw the base first, then swing compass arcs of the two remaining lengths from its endpoints; where the arcs cross is the third vertex.
Leave the construction arcs visible - exam mark schemes require them.
A net is a flat arrangement of faces that folds into a solid; know the nets of cubes, cuboids, prisms and pyramids.
Measurements from a net give surface area (total area of the faces) and feed volume calculations for prisms.
Scale drawings and bearings C4.3 / E4.3
A scale of 1 : n means 1 unit on the drawing stands for n units in real life; multiply drawing lengths by n, and convert units at the end (cm to m to km).
Three-figure bearings are measured clockwise from north and always written with three digits: 007°, 065°, 240°.
A back bearing differs by 180°: if the bearing of B from A is 065°, the bearing of A from B is 065° + 180° = 245°.
Mark the north line at the point you are measuring from - bearings are always 'from' somewhere, and the north line belongs at that point.
Similarity C4.4 / E4.4
Similar shapes have equal angles and matching sides in the same ratio; the scale factor is new length / old length.
Find an unknown side by pairing it with its matching side and applying the scale factor - draw the two shapes separately if they overlap.
Extended: areas of similar shapes scale by (length factor)^2 and volumes of similar solids by (length factor)^3.
Extended: show two triangles are similar by matching equal angles (AA is enough) or showing all three side ratios are equal.
Symmetry C4.5 / E4.5
Line symmetry: a fold line that maps the shape onto itself. A rectangle has 2, a square 4, a regular pentagon 5.
Rotational symmetry: the order is how many times the shape fits onto itself in one full turn. A rectangle has order 2, a square order 4, a regular hexagon order 6.
A regular polygon with n sides has n lines of symmetry and rotational symmetry of order n.
Extended: solids can have planes of symmetry (a cuboid has 3) and axes of symmetry.
Angle facts and parallel lines C4.6 / E4.6
Angles at a point sum to 360°; angles on a straight line sum to 180°; vertically opposite angles are equal.
Angles in a triangle sum to 180° and in a quadrilateral to 360°; an exterior angle of a triangle equals the sum of the two opposite interior angles.
With parallel lines and a transversal: corresponding angles are equal, alternate angles are equal, and co-interior angles sum to 180°.
Give the reason with each step of an angle chase - 'alternate angles are equal' earns the mark as much as the number.
Regular polygons: each exterior angle is 360° / n, each interior angle is 180° - 360° / n, and the interior angles sum to (n - 2) x 180°.
Circle theorems I C4.7 / E4.7
The angle in a semicircle is 90°: any angle standing on a diameter is a right angle.
The angle between a tangent and the radius at the point of contact is 90°.
Extended: the angle at the centre is twice the angle at the circumference standing on the same arc.
Extended: angles in the same segment (standing on the same chord, on the same side) are equal.
Extended: opposite angles of a cyclic quadrilateral sum to 180°.
Extended: the alternate segment theorem - the angle between a tangent and a chord equals the angle in the alternate segment.
Circle theorems II (Extended) E4.8
Equal chords are equidistant from the centre of the circle.
The perpendicular bisector of a chord passes through the centre - so the line from the centre to the midpoint of a chord meets it at 90°.
Tangents drawn to a circle from the same external point are equal in length.
These are symmetry facts: folding the circle along the diameter through the midpoint of a chord maps the figure onto itself.
EduTribe study notes - original material aligned to the Cambridge IGCSE Mathematics 0580 syllabus (2025-2027). Automated checks completed; not yet reviewed by an educator.