Geometry

• Fact → Reason → Link. State the fact you are using, name the exact property that licenses it, then link it to the quantity you were asked for. Three short lines beat one long sentence that names nothing.
• C4.2 and E4.2 are word-for-word identical in the official tables. Everything in this section is required on both routes, and there is no Extended depth panel to look for.
• C4.3 and E4.3 are word-for-word identical in the official tables. Everything in this section is required on both routes, and there is no Extended depth panel to look for.
• At any one vertex, the interior angle and the exterior angle lie on a straight line, so they always add to \(180^\circ\). That single fact converts either formula into the other, so you only need to remember one of them.
• There are only two. If a Core circle question is not opening, the thing you have not found is either a diameter or a tangent — there is no third possibility to look for.
• Properties 7, 8 and 9 hand you an equal length or a right angle; properties 3 to 6 hand you an angle. That is why a hard circle question usually starts in E4.8 and finishes in E4.7.

• Use point, vertex, line, parallel, perpendicular and bearing correctly.
• Classify an angle as acute, right, obtuse or reflex, and identify interior and exterior angles of a polygon.
• Explain what similar, congruent and scale factor mean.
• State the defining properties of equilateral, isosceles, scalene and right-angled triangles.
• State the defining properties of the square, rectangle, kite, rhombus, parallelogram and trapezium.
• Name regular and irregular polygons including the pentagon, hexagon, octagon and decagon.
• Name the simple solids — cube, cuboid, prism, cylinder, pyramid, cone and sphere — and use face, surface and edge.
• Name every circle part on the Core list: centre, radius, diameter, circumference, semicircle, chord, tangent, arc, sector and segment.
• Measure and draw lines and angles, ruling every straight edge.
• Construct a triangle given all three sides using only a ruler and a pair of compasses, leaving the construction arcs visible.
• Construct a rhombus by drawing two such triangles on opposite sides of a diagonal.
• Decide whether three given lengths can form a triangle at all.
• Draw, read and use a net, and use measurements from a net to find a surface area or a volume.
• Read a ratio scale and convert between drawing length and real length in both directions.
• Interpret and construct scale drawings, using a ruler for every straight edge.
• Write, measure and draw three-figure bearings clockwise from a north line drawn at the correct point.
• Use the terms north, east, south and west, and calculate a reverse bearing.
• Identify corresponding vertices, angles and sides in similar figures in the right order.
• Calculate an unknown length in a pair of similar shapes from a length scale factor.
• Count lines of symmetry and the order of rotational symmetry for triangles, quadrilaterals and regular polygons.
• Use the symmetry properties of a shape to justify a statement about its sides or angles.
• Use angles at a point, angles on a straight line, vertically opposite angles and the triangle and quadrilateral angle sums.
• Use corresponding, alternate and co-interior angles on parallel lines, naming each one exactly.
• Use the interior-angle sum \((n-2)\times180^\circ\) and the exterior-angle sum \(360^\circ\), and find each angle of a regular polygon.
• Read and write three-letter angle notation such as \(\angle ABC\).
• State and apply the angle in a semicircle: an angle subtended at the circumference by a diameter is \(90^\circ\).
• State and apply the tangent–radius right angle at the point of contact.
• Build a short chain from those two properties in which every step carries its own exact reason.
• Use plane and perpendicular bisector correctly.
• Name the hemisphere and the frustum as well as the simple solids.
• Distinguish the major arc from the minor arc.
• Move between length, area, surface-area and volume ratios using \(k\), \(k^2\) and \(k^3\) in both directions.
• Show that two triangles are similar by giving geometric reasons.
• Solve problems and give simple explanations involving similarity.
• Recognise the symmetry properties of prisms, cylinders, pyramids and cones.
• Identify planes of symmetry and axes of rotational symmetry of a solid, and keep them apart from lines of symmetry and centres of rotation.
• Use angle properties of irregular polygons as well as regular ones.
• State and apply the angle at the centre being twice the angle at the circumference on the same arc.
• State and apply the equality of angles in the same segment.
• State and apply the cyclic quadrilateral property.
• State and apply the alternate segment theorem.
• Use the fact that equal chords are equidistant from the centre, and its converse.
• Use the fact that the perpendicular bisector of a chord passes through the centre.
• Use the fact that tangents from an external point are equal in length.
• Build a multi-step chain that combines E4.7 and E4.8 properties, justifying every step separately.

Where this shows up outside an exam. Bearings and scale drawings are how navigation charts, hiking maps and flight plans are actually read. Similarity and the \(k^2\), \(k^3\) rules are why a scale model of a building needs far less paint but disproportionately less material, and why doubling a pipe’s diameter roughly quadruples the water it can carry. Circle theorems underpin the geometry of gears, cams and any linkage turning about a fixed centre.

Similarity

Two figures are similar when their corresponding angles are equal and their corresponding sides are all in the same ratio. That common ratio is the length scale factor k, and calculating lengths from it is the whole of the Core requirement C4.4. Extended E4.4 adds that areas and surface areas of similar figures are in the ratio k squared and volumes are in the ratio k cubed, so a length ratio is recovered from an area ratio by taking a square root and from a volume ratio by taking a cube root.

Scale Drawings

A scale drawing represents a real object or route at a fixed ratio, so that every drawn length is the corresponding real length divided by the scale factor. The scale is written as a ratio such as 1 to 50 000, in which both sides are in the same unit. Directions on such a drawing are given as three-figure bearings, measured clockwise from a north line drawn at the point the bearing is measured from.

Geometrical Terms

The agreed vocabulary of Cambridge IGCSE Mathematics 0580 geometry: the names and defining properties of lines, angles, triangles, quadrilaterals, polygons, solids and the parts of a circle, listed separately for the Core route (C4.1) and the Extended route (E4.1). A defining property is the minimum condition a shape must satisfy to be given that name, and it is the only thing a name entitles you to assume about a figure.

Symmetry

A shape has symmetry when a reflection or a rotation leaves it looking exactly as it did before. In two dimensions a line of symmetry is a mirror line across which the shape maps onto itself, and the order of rotational symmetry is the number of times the shape matches itself during one full turn of 360 degrees. In three dimensions a plane of symmetry cuts a solid into mirror-image halves, and an axis of rotational symmetry is a line about which the solid can be turned onto itself.

Circle Theorems I

The angle properties of a circle required by Cambridge IGCSE Mathematics 0580. Core C4.7 requires the first two: an angle subtended by a diameter at the circumference is a right angle, and a tangent is perpendicular to the radius at the point of contact. Extended E4.7 requires those two and four more: the angle at the centre is twice the angle at the circumference when both stand on the same arc; angles at the circumference standing on the same arc or chord are equal; opposite angles of a cyclic quadrilateral sum to 180 degrees; and the angle between a tangent and a chord equals the angle in the alternate segment.

Geometrical Constructions

Accurate drawing with mathematical instruments: measuring and drawing lines to the nearest millimetre and angles to the nearest degree, and constructing a triangle from three given side lengths using only a ruler and a pair of compasses, with the compass arcs left visible as evidence. Also covers nets, the flat patterns that fold to make a solid without gaps or overlaps.

Circle Theorems II

The three symmetry properties of a circle required by Cambridge IGCSE Mathematics 0580 Extended candidates only, under E4.8: chords of equal length are the same perpendicular distance from the centre; the perpendicular bisector of a chord passes through the centre, so the perpendicular dropped from the centre to a chord bisects it; and the two tangents drawn to a circle from the same external point are equal in length. All three follow from the circle's reflection symmetry rather than from any angle rule.

Angles

The angle facts that Cambridge IGCSE Mathematics 0580 requires, in identical wording on both routes: angles at a point sum to 360 degrees, angles on a straight line sum to 180 degrees, vertically opposite angles are equal, the angles of a triangle sum to 180 degrees and those of a quadrilateral to 360 degrees. Where a transversal crosses parallel lines, corresponding angles are equal, alternate angles are equal and co-interior angles sum to 180 degrees. For an n-sided polygon the interior angles sum to n minus two lots of 180 degrees while the exterior angles always sum to 360 degrees; Core applies this to regular polygons and Extended applies it to regular and irregular polygons alike.

• “It looks like a right angle, so I will use \(90^\circ\).” Core C4.6 Fix A diagram gives you exactly three things: stated values, drawn markings, and whatever follows from a named property. Nothing else.
• “My reason was ‘circle theorem’.” Core C4.7 Fix Name the property in full and identify the arc, chord or tangent it applies to. Core has two circle properties to choose between and Extended has nine; saying which one is what makes the explanation complete.
• “Co-interior angles are equal.” Core C4.6 Fix They sum to \(180^\circ\). Corresponding and alternate angles are the equal ones. Co-interior angles are supplementary, and they are the pair students most often name wrongly.
• “The bearing is \(25^\circ\).” Core C4.3 Fix Bearings are written with three figures: \(025^\circ\). Also check you measured clockwise, from a north line drawn at the point you are measuring from.
• “A sector and a segment are much the same thing.” Core C4.1 Fix A sector is bounded by two radii and an arc. A segment is bounded by a chord and an arc. The word chosen decides which region is meant.
• “I rubbed out the compass arcs to make it neat.” Core C4.2 Fix The arcs are the evidence of construction, and the syllabus requires them to be shown. Leave them. A neat triangle with no arcs looks exactly like a measured guess.
• “Every polygon angle question uses \(360^\circ \div n\).” Core C4.6 Fix That formula gives one exterior angle of a regular polygon. Core polygon questions are about regular polygons, so it usually applies — but check the word regular is actually there before you divide.
• “The areas are in the ratio \(49:81\), so the lengths are too.” Extended E4.4 Fix Take the square root: lengths are \(7:9\). For volumes, take the cube root. Going the wrong way through \(k^2\) and \(k^3\) is the most common similarity error there is. Core similarity never uses \(k^2\) or \(k^3\) at all.
• “The tangent is perpendicular to the chord.” Core C4.7 Extended E4.7 Fix A tangent is perpendicular to the radius at the point of contact — that is the Core property. On Extended, its angle to a chord is governed by the alternate segment theorem instead, and is generally not \(90^\circ\).
• “A prism has four planes of symmetry.” Extended E4.5 Fix Count from the cross-section you are given. An equilateral triangular prism has \(4\); a scalene one has \(1\). Three-dimensional symmetry is an Extended requirement; Core symmetry is counted in two dimensions only.
• Measuring a diagram instead of reasoning from it Core C4.1 Fix Use the stated values, the markings on the diagram, and named properties. A diagram is a reasoning aid, not a measuring instrument, unless the question explicitly asks you to measure or construct. Why it matters The syllabus asks for the property to be named, so an answer produced by measuring has not shown the reasoning being assessed — and since exam diagrams are not drawn to scale, the measurement is usually wrong as well.
• Erasing the construction arcs to make the drawing tidy Core C4.2 Fix Leave every compass arc on the page. They are part of the answer. Why it matters Without the arcs, a constructed triangle is indistinguishable from a measured or estimated one.
• Writing a bearing with two digits, e.g. \(25^\circ\) Core C4.3 Fix Three figures always: \(025^\circ\). Check as well that you measured clockwise, from a north line drawn at the point you are measuring from. Why it matters Three-figure notation is part of the skill being assessed, not a presentation preference.
• Using an area ratio directly as a length ratio Extended E4.4 Fix Take the square root of an area ratio, and the cube root of a volume ratio, to get back to lengths. Areas \(49:81\) give lengths \(7:9\). Why it matters Every later value in the question is built on the length ratio, so the error carries forward into all of them rather than staying in one line.
• Scaling a surface area with \(k^3\) because the object is a solid Extended E4.4 Fix Surface area is an area: it scales with \(k^2\). Only volume and capacity use \(k^3\).
• “Co-interior angles are equal” Core C4.6 Fix Co-interior angles sum to \(180^\circ\). The equal pairs are corresponding and alternate angles. Why it matters A correct number with the wrong property named is not a correct explanation, and the explanation is what the syllabus asks for.
• Giving “circle theorem” as the reason Core C4.7 Fix Name the property and say which arc, chord, diameter or tangent it applies to. On Core there are two properties to choose between; on Extended there are nine, so “angles in the same segment are equal, both standing on chord \(AB\)” is the level of detail required.
• Doubling or halving on the wrong arc Extended E4.7 Fix The angle at the centre is twice the angle at the circumference on the same arc. If the point at the circumference is on the minor arc, its angle stands on the major arc, whose central angle is \(360^\circ - \theta\).
• Confusing a sector with a segment Core C4.1 Fix Two radii make a sector; a chord makes a segment. The word in the question decides which region is meant.
• “The tangent is perpendicular to the chord” Core C4.7 Extended E4.7 Fix The tangent is perpendicular to the radius at the point of contact. Its angle to a chord is given by the alternate segment theorem.
• Assuming a quadrilateral is a parallelogram because two sides look parallel Core C4.1 Fix Check the defining property against what is given or marked. One pair of parallel sides makes a trapezium, not a parallelogram.
• Quoting “a prism has 4 planes of symmetry” from memory Extended E4.5 Fix Count from the given cross-section. An equilateral triangular prism has \(4\); a scalene one has \(1\).
• Dividing before converting in a scale-drawing question Core C4.3 Fix Convert the real length into the drawing unit first, then divide by the scale factor. Both sides of a ratio scale are in the same unit.
• Using the regular-polygon formulas on an irregular polygon Extended E4.6 Fix \(\frac{360^\circ}{n}\) is the exterior angle of a regular polygon only. For an irregular polygon you have the two sums — \((n-2)\times180^\circ\) and \(360^\circ\) — and nothing about individual angles.
• Rounding part-way through an angle chain Core C4.6 Fix Keep exact values throughout. If an angle is \(67.5^\circ\), carry \(67.5^\circ\), not \(68^\circ\), unless a degree of accuracy has been requested.

• The single most useful line in this table is 4.7. A Core candidate who memorises six circle theorems has spent revision time on four properties they will not be asked about; an Extended candidate who learns only two will be unable to start most circle questions. Check which route you are entered for before you revise this chapter.
• Work with a ruler, a protractor and a pair of compasses beside you. Sections 4.2 and 4.3 contain instructions you cannot learn by reading, and both routes are assessed on them identically. Draw each construction once as you meet it; it takes four minutes and it is the difference between a construction mark and a blank page.
• Read the diagram like an examiner. A tick mark means equal lengths. A double arc means equal angles. Arrowheads mean parallel. A small square means \(90^\circ\). Anything not marked and not given is not available to you, however convincing it looks.
• Self-test honestly. Tick a box only if you could produce the result on blank paper with the correct reason written beside it. “I would recognise it” is not the same skill and is not what is assessed.
• Read your own row, not the other one. Core is assessed on C4.1 to C4.7; a Core paper cannot ask for an alternate-segment reason or an equal-tangent length. Extended is assessed on E4.1 to E4.8, and the Extended circle work is roughly four times the size of the Core circle work. Nothing else in Topic 4 is reserved for one paper within a route, and no theorem is guaranteed to appear — prepare your whole route rather than betting on a subset.
• Instruments and accuracy, stated by the syllabus. A pair of compasses, a protractor and a ruler are expected in every paper, and tracing paper may be requested. On the calculator papers, give a non-exact numerical answer to three significant figures and an angle in degrees to one decimal place, unless the question says otherwise — and do not round intermediate values on the way there. An answer that is exact stays exact: if an angle works out to \(67.5^\circ\), write \(67.5^\circ\), not \(68^\circ\).
• A practical habit. Answer in two columns on your own paper: the statement on the left, the reason on the right. It makes an omitted reason visible to you before it becomes visible to a marker.
• Two reasons in one step is a warning sign. If a single line of your working needs “alternate angles and the triangle sum”, split it into two lines. Chains lose marks by being compressed far more often than by being long.
• Do not estimate a value from a diagram in a reasoning question. Measuring is the correct method only where the question explicitly asks you to measure or to construct accurately. Everywhere else it is a wrong method.
• Drawing a bearing accurately. Draw the north line first, put the protractor centre on the point with its base along the north line, count clockwise, mark the angle, then rule the line. If the bearing is over \(180^\circ\), measure the clockwise angle in two stages — \(180^\circ\) first, then the rest — rather than trying to read a reflex angle from one setting.
• The rhombus and the rectangle catch people out. A rhombus’s lines of symmetry are its two diagonals; a rectangle’s are the two lines through the midpoints of opposite sides. Neither shape has all four, and swapping them over is a common error. Only the square, which is both, has four.
• Core candidates: this is where to stop, and that is not a shortcut. If a Core question involves a circle, the answer runs through a diameter or a tangent — there is no third possibility. So the first question to ask of a Core circle diagram is: does a chord pass through the centre, or does a line touch the circle once? One of the two will be there.
• How to spot which of the three you need. If the question mentions two chords, or gives you two lengths that look like chord distances, reach for property 1. If a right angle from the centre meets a chord, or a midpoint is marked, reach for property 2. If two tangents come from one outside point, reach for property 3 — and expect an isosceles triangle to appear immediately afterwards.
• “Not drawn accurately” is an instruction, not a disclaimer. It tells you the lengths and angles on the page are not to scale, so a measured answer will usually be wrong as well as unearned. Every diagram in this chapter carries the same badge for the same reason. Measuring is the correct method only where the question explicitly says measure, construct, or draw accurately.
• Last thirty seconds of a geometry question. Read your own working back and ask of every line: does this line name a property? If a line has a number and no property, add the property. That is usually the difference between a partial explanation and a complete one.
• The Extended-specific check. Before writing a circle reason, ask which of the nine you are quoting and which arc, chord, diameter or tangent it applies to. Nine candidate reasons is enough that a vague one identifies nothing — and the angle-at-the-centre property gives two different answers depending on which arc you picked.
• Do not spend the hour re-reading the diagrams. They will feel familiar, which feels like progress and is not. Recall is what transfers; recognition is not.