Geometry

The marks here are for naming rules, not finding numbers.

Mathematics · 0580 Extended Topic 7 of 12

Basic angles, triangles and quadrilaterals

FIG 4.0 Each leg of a journey is read clockwise from its own north line.

Nearly every geometry mark rests on a handful of facts about angles meeting at a point or inside a simple shape.

Angles that meet

Angles round a full turn sum to 360°, angles on a straight line sum to 180°, and where two lines cross the vertically opposite angles are equal.

Definition

Vertex, reflex angle, vertically opposite

A vertex is the point at which two or more lines or edges meet (plural: vertices). A reflex angle is greater than 180° and less than 360°. Vertically opposite angles are the equal pair formed opposite one another where two straight lines cross.

FIG 4.1 The three ways straight lines meet, and the fact each one gives you.

Angle sums in triangles and quadrilaterals

The angles of any triangle sum to 180°, and those of any quadrilateral to 360°. A diagonal cuts a quadrilateral into two triangles, which is where the second result comes from. Neither depends on the shape being regular.

FIG 4.2 A diagonal splits any quadrilateral into two triangles.

Worked example: angles of 82° and 148° sit alongside angle ACB at C. In triangle ABC, AC = BC. Angles at a point sum to 360°, so angle ACB = 360 − 148 − 82 = 130°. AC = BC makes the triangle isosceles, so angles CAB and CBA are equal: angle CAB = (180 − 130) ÷ 2 = 25°. Two facts had to be chained — most Extended angle questions are chains.

Examiner note

When a question says "giving a reason", the rule must be named: "angles on a straight line sum to 180°". Describing what was done — "I took it away from 180" — earns the value but not the reason.

Why this matters

Roof trusses and bridge frames are built from triangles because a triangle is the only polygon whose angles are fixed once its sides are — it cannot be pushed out of shape.

Angles in polygons

Interior angles are not memorised shape by shape: any polygon cuts into triangles, and each triangle contributes 180°.

Definition

Polygon, interior and exterior angle

A polygon is a closed two-dimensional shape with straight sides — regular when all sides and angles are equal, irregular otherwise. At each vertex the interior angle sits inside the shape and the exterior angle continues the side outwards; they lie on a straight line, so they sum to 180°.

The sum of the interior angles

Join one vertex of an n-sided polygon to every non-adjacent vertex and it splits into exactly n − 2 triangles. This holds for irregular polygons as well as regular ones: sum of interior angles = 180° × (n − 2).

Regular polygons

In a regular polygon every angle is identical, so the interior sum is simply shared out: interior angle = 180°(n − 2) ÷ n, and exterior angle = 360° ÷ n. The exterior angles always total 360°, which is usually the faster route.

FIG 4.3 Interior angle sum by triangulation; exterior angles round the perimeter.

Worked example: the interior angle of a regular polygon with 20 sides. Exterior angles sum to 360°, so each one is 360 ÷ 20 = 18°. Interior and exterior angles lie on a straight line, so the interior angle is 180 − 18 = 162°. Checking directly: 180(20 − 2) ÷ 20 = 3240 ÷ 20 = 162°.

Examiner note

Mixing up 180(n − 2) ÷ n with 360 ÷ n is the most common polygon error. Read whether the question wants the interior or the exterior angle before starting.

Why this matters

Only triangles, squares and regular hexagons tile a flat surface without gaps, because only their interior angles divide exactly into 360°. Honeycomb, floor tiles and hex bolts all follow from that.

Angles in parallel lines

When a straight line crosses a pair of parallel lines, the eight angles it creates take only two different values. Three named rules tell you which angles share a value and which pair up to 180°.

Definition

Parallel lines and transversal

Parallel lines lie in the same plane and never meet, however far they are extended. A transversal is a line that crosses two or more other lines.

Three rules, one transversal

Corresponding angles are equal — same side of the transversal, same side of each parallel line. Alternate angles are equal — opposite sides of the transversal, both between the parallel lines. Co-interior angles sum to 180° — same side of the transversal, both between the parallel lines.

FIG 4.4 Corresponding, alternate and co-interior angles.

Worked example: AB is parallel to CD. The angle inside and right of the transversal at AB is 62°. Angle y is co-interior with it at CD; angle z is vertically opposite y. y = 180 − 62 = 118°, because co-interior angles sum to 180°. z = y = 118°, because vertically opposite angles are equal. Each value needed its own named rule — the names carry as many marks as the numbers.

Examiner note

Never write "F-angle", "Z-angle" or "C-angle" as a reason. The mark scheme credits only corresponding, alternate and co-interior; the letter shapes are a way of spotting them, not a way of naming them.

Why this matters

Technical drawing depends on these rules to carry an angle from one parallel ruling to another without measuring it a second time — and so accumulating error.

Symmetry in two and three dimensions

Symmetry is described two ways at Extended: by reflection, which counts mirror lines and mirror planes, and by rotation, which counts how often a shape fits onto itself.

Definition

Line, order and plane of symmetry

A line of symmetry is a mirror line: folding the shape along it maps each half exactly onto the other. The order of rotational symmetry is the number of positions in one full turn at which the shape looks unchanged. A plane of symmetry is a flat surface cutting a solid into two mirror-image halves (Extended).

Symmetry in two dimensions

A 2D shape is described by its lines of symmetry and its order of rotational symmetry. The two are independent: a parallelogram has no lines at all (0 lines, order 2), yet an equilateral triangle has 3 lines and order 3, and a regular pentagon has 5 lines and order 5.

FIG 4.5 Lines of symmetry and order of rotation, counted separately.

ExtendedSymmetry in three dimensions

A solid is described by its planes of symmetry and its axes of rotational symmetry. A cuboid with unequal edges has 3 planes; an isosceles triangular prism has 2 planes; a cylinder has infinitely many upright planes plus one horizontal, and 1 axis; a cone has infinitely many upright planes, none horizontal, and 1 axis; a square-based pyramid has 4 planes and 1 axis, of order 4.

FIG 4.6 A plane cuts a solid into mirror halves; an axis is a line it turns about.

Examiner note

Order of rotational symmetry is never zero. A shape with no rotational symmetry has order 1, because it returns to itself after a full turn — writing 0 loses the mark.

Why this matters

Packaging designers count a carton's planes of symmetry to work out the smallest number of distinct printing plates the design needs.

Constructions and nets

Cambridge asks for constructions with ruler and compasses only. The arcs are the evidence of method, and a protractor leaves none.

Definition

Perpendicular bisector and net

A perpendicular bisector is the line cutting a segment exactly in half at a right angle, drawn with equal arcs struck from both ends (Extended). A net is a two-dimensional arrangement of faces that folds up to form a solid, with no face overlapping another.

Constructing a triangle from three sides

The third vertex is the one point at the correct distance from both ends of the base. Each arc records every point at one of those distances; where they cross is the vertex.

Worked example: construct triangle ABC with AB = 8 cm, AC = 7 cm, BC = 5 cm. Draw AB = 8 cm with a ruler and label the ends A and B. Open the compasses to 7 cm and arc from A. Open them to 5 cm, arc from B, and label the crossing point C. Join AC and BC, leaving both arcs clearly visible. Three sides fix a triangle completely — no angle was ever measured.

FIG 4.7 Construct a triangle from three sides, arcs left on the page.

Nets

A net shows every face laid flat. A solid usually has several valid nets; each face must appear exactly once, with equal edges where they meet on folding.

FIG 4.8 Six faces, each once, with matching edges where they will meet.

Examiner note

Construction arcs must be left on the page. Examiners mark the method from the arcs, so erasing them costs the method mark even when the finished triangle measures perfectly.

Why this matters

A ruler-and-compass construction is exact by geometry rather than by measurement — the same logic that modern CAD software uses when it solves a drawing from its constraints.

Scale drawings and bearings

A bearing answers one question exactly: standing here, in which direction is that? Measured always clockwise from north in three figures, it cannot be misread.

Definition

Bearing and scale

A bearing is a direction measured clockwise from north, written with three figures from 000° to 360°. A scale is the ratio of a length on the drawing to the true length, written 1 : n with both lengths in the same units.

Three-figure bearings

A bearing is measured from the north line where you stand, turning clockwise to face the destination. Each end of a journey has its own north line, so forward and back bearings differ by 180°.

FIG 4.9 The same journey read from each end; a back bearing differs by 180°.

Scale drawings

A scale of 1 : n means every drawn length stands for n of the same units in reality. Multiply to go to reality, divide to come back, and convert units last.

FIG 4.10 One scale key converts every drawn length at the same rate.

Worked example: a drawing with AB = 8 cm shows a field at scale 1 : 10 000, with B due east of A. Actual AB = 8 × 10 000 = 80 000 cm = 800 m = 0.8 km. B is due east of A, so A is due west of B; due west is three quarters of a turn clockwise from north, so the bearing of A from B is 270°.

Examiner note

Three-figure bearings always carry their leading zeros: write 059°, never 59°. The angle is right but the accuracy mark is lost.

Why this matters

Marine and air navigation quote headings as three figures for the same reason: over a radio, 059 and 590 can never be mistaken for one another.

Similarity — lengths

FIG 4.0 Cyclic quadrilateral and angle at the centre — two of the theorems this half of the chapter builds toward.

Similarity is the one genuinely computational idea in this half of the chapter. Everything else here is a theorem to recall; this is a number to find, and then to raise to a power.

Similar shapes

Two shapes are similar when one is an enlargement of the other. Equal angles alone are enough for triangles, and the ratio between any pair of corresponding sides gives the scale factor k: k = new length ÷ original length.

Definition

Similar, congruent, scale factor

Similar shapes are an enlargement of one another: corresponding angles are equal and corresponding sides are all in the same ratio. Congruent shapes are the same shape and size — Cambridge asks you to recognise congruence only, never to prove it. The scale factor k is the number that multiplies every length on the first shape to give the matching length on the second.

Showing two triangles are similar

FIG 4.1 Similar triangles, and showing similarity from two equal angles.

Extended candidates are asked to show similarity, not just use it. Two equal angles are sufficient, and each one must be justified by a named rule from earlier in this chapter or by a circle theorem.

Worked example: triangle CXD is similar to triangle BXA. DX = 8.0 cm, AX = 4.0 cm, BX = 2.7 cm. The naming order pairs CX with BX, and XD with XA. Scale factor k = XD ÷ XA = 8.0 ÷ 4.0 = 2. CX = k × BX = 2 × 2.7 = 5.4 cm.

Examiner note

Read the two triangles in matching order. "Triangle CXD is similar to triangle BXA" pairs C with B, X with X and D with A. Most lost marks here come from pairing the wrong sides.

Why this matters

Every scale model, map and architectural plan is a similar figure. The scale factor is the single number connecting the drawing to the building.

Similarity — areas and volumes

One scale factor governs a pair of similar solids, but it acts on lengths, areas and volumes at three different powers.

Definition

Area and volume scale factor

When every length scales by k, area scales by k² and volume by k³ — the number multiplying every area or volume of a similar shape or solid.

Why the powers change

An area is a product of two lengths, so multiplying every length by k multiplies every area by k × k. A volume is a product of three lengths, so it is multiplied by k × k × k: area ratio = k², volume ratio = k³. Surface areas scale like areas.

FIG 4.2 Doubling every length quadruples every area and multiplies every volume by eight.

Worked example: two similar cones have volumes 32 cm³ and 108 cm³. The smaller cone is 10 cm high. Volume ratio = 108 ÷ 32 = 27 ÷ 8 = k³, so k = ∛(27/8) = 3 ÷ 2 = 1.5. Height of the larger cone = 1.5 × 10 = 15 cm. Surface areas are in the ratio k² = 1.5² = 2.25, i.e. 4 : 9.

Examiner note

Read which ratio the question gives you. If it hands you an area or a volume ratio, take the square or cube root before touching any length.

Why this matters

Doubling a shipping container's length multiplies its steel area by four but its capacity by eight — which is why the largest vessels carry cargo most cheaply.

Circle theorems — centre and semicircle

Three theorems carry most of the circle work in the Extended papers. Each is a sentence to recall exactly, because the mark scheme pays for the sentence as well as the number.

The three foundation theorems

Angle at the centre: the angle at the centre is twice the angle at the circumference — both angles must stand on the same arc. Angle in a semicircle: the angle in a semicircle is a right angle — a special case of the first theorem, where the angle at the centre is the straight 180° of the diameter. Tangent and radius: a tangent meets the radius at 90°, at the point of contact.

Definition

Chord, tangent, major and minor arc

A chord is a straight line joining two points on the circumference; a diameter is the longest chord. A tangent is a straight line touching the circle at exactly one point, the point of contact. Two points divide the circumference into a longer major arc and a shorter minor arc (Extended).

FIG 4.3 Angle at the centre, angle in a semicircle, tangent and radius.

Worked example: A, B and C lie on a circle, centre O. Angle AOB = 108° and C lies on the major arc AB. Angle AOB and angle ACB both stand on the minor arc AB, so angle ACB = 108 ÷ 2 = 54°, because the angle at the centre is twice the angle at the circumference. Had C been on the minor arc, the relevant angle at the centre would be the reflex 252°, giving 126° instead.

Examiner note

When the angle at the centre looks obtuse or reflex, decide which of the two angles at O the question actually wants before doubling or halving anything.

Why this matters

Structural arches carry load along the line of the radius, so engineers use the tangent–radius right angle to work out where the thrust leaves the curve.

Circle theorems — segments

The next three theorems all describe angles standing on a chord. Two of them are equalities, one is a sum — and mixing those up is the quickest way to lose both marks at once.

Definition

Segment and cyclic quadrilateral

A segment is the region between a chord and one of the two arcs it cuts off — a major and a minor segment. A cyclic quadrilateral is a quadrilateral whose four vertices all lie on the circumference of one circle.

Three theorems about segments

Same segment: angles in the same segment, standing on the same arc, are equal (both angles must be drawn from the ends of the same chord). Cyclic quadrilateral: opposite angles of a cyclic quadrilateral sum to 180° (check all four vertices really are on the circle first). Alternate segment: the angle between a tangent and a chord equals the angle in the alternate segment — the segment on the far side of the chord from the given angle.

FIG 4.4 Same segment, cyclic quadrilateral, alternate segment.

Worked example: TA is a tangent to a circle at A. Chord AB makes an angle of 58° with the tangent, and C lies on the major arc AB. Angle ACB lies in the alternate segment, so by the alternate segment theorem angle ACB = 58°. "Alternate segment theorem" is the exact wording credited; describing the picture earns nothing.

Examiner note

"They look equal" scores zero. Quote the theorem by name: angles in the same segment are equal; opposite angles of a cyclic quadrilateral sum to 180°.

Why this matters

Ships once fixed their position from the angle between two landmarks: every point giving the same angle lies on one arc, which is the same-segment theorem read backwards.

Circle theorems — chords and tangents

The last three properties are Extended-only and have no Core equivalent at all. Each one comes from the circle’s symmetry rather than from an angle chase.

Definition

Equidistant and point of contact

Equidistant means the same distance from — the distance from the centre to a chord is always measured along the perpendicular. The point of contact is the single point at which a tangent touches the circle; two tangents from one external point give two of them.

ExtendedThree symmetry properties

Equal chords: equal chords are the same distance from the centre — and the converse, chords equidistant from the centre are equal. Perpendicular bisector of a chord: the perpendicular bisector of a chord passes through the centre — read backwards, a radius at right angles to a chord bisects it. Tangents from an external point: two tangents drawn from the same external point are equal in length — the two radii and two tangents form a kite.

FIG 4.5 Equal chords, bisector of a chord, equal tangents.

Worked example: A, B and C lie on a circle, centre O. TA and TB are tangents. Angle ACB = 67°. By the alternate segment theorem, angle TAB = angle ACB = 67°. TA = TB (tangents from an external point are equal), so triangle TAB is isosceles and angle TBA = 67° too. Angle ATB = 180 − 67 − 67 = 46°. A second route scores equally: angle AOB = 2 × 67 = 134°, and the kite OATB gives 360 − 90 − 90 − 134 = 46°.

Examiner note

More than one route is usually accepted. Cambridge credits any valid chain of geometry, so use whichever facts you can see fastest rather than hunting for the "intended" method.

Why this matters

These three properties describe how a straight line meets a curve — the same geometry that shapes satellite dishes and parabolic reflectors.

Exam advice

Common mistakes

Writing "F-angle", "Z-angle" or "C-angle" as the reason

The letter shapes are a way of spotting the pair, not their names. The reason mark is lost even when the number is correct.

Erasing the construction arcs once the triangle is drawn

Examiners mark the method from the arcs, so a perfectly accurate triangle with no arcs still loses the method mark.

Giving a bearing with fewer than three figures

Writing 70° instead of 070° loses the accuracy mark, however correct the angle itself is.

Confusing the interior and exterior angle formulas

Using 360 ÷ n where 180(n − 2) ÷ n was needed produces the exterior angle and scores nothing for an interior-angle question.

Comparing the angle at the centre with the wrong arc

Decide which arc the centre angle stands on before doubling or halving.

Giving no reason, or an imprecise one

"They look the same" scores nothing; only the exact theorem name earns the mark.

Trying to prove two shapes are congruent

Outside the Extended syllabus, which asks for recognition of congruence only.

Mixing up the length, area and volume scale factors

Applying k where k² or k³ was needed is the commonest similarity error.

Model answer

A, B, C, D and E lie on a circle. AC and BD intersect at X. Angle ACD = 55° and angle CXD = 88°. Find (i) angle CDB, (ii) angle ABD, (iii) angle AED, giving a geometrical reason for each answer.

[6 marks]

Angle CDB = 37°

In triangle CXD: 180 − 88 − 55 = 37°.

Reason: the angles of a triangle sum to 180°

The value on its own earns only half of this part.

Angle ABD = 55°

Angle ABD and angle ACD both stand on arc AD.

Reason: angles in the same segment are equal

Pointing at the segment is not enough — name the theorem.

Angle AED = 125°

180 − 55, using cyclic quadrilateral ABDE.

Reason: opposite angles of a cyclic quadrilateral sum to 180°

ABDE has all four vertices on the circle.

Recall checklist

State the three basic angle facts.
Calculate a missing angle in a triangle or quadrilateral.
Calculate the interior angle sum for n sides.
Calculate both angles of a regular polygon.
Distinguish the three parallel-line rules.
State lines and order of symmetry for a 2D shape.
Construct a triangle from three sides, arcs shown.
Calculate a bearing from a scale drawing.
Find a missing length using the length scale factor.
Apply k, k² and k³ to similar shapes and solids.
Show two triangles are similar, with a reason per angle.
State the angle at the centre theorem.
State the semicircle and tangent–radius theorems.
State the same segment and cyclic quadrilateral theorems.
Apply the alternate segment theorem.
State the two chord theorems and the tangent-length rule.

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Geometry

Basic angles, triangles and quadrilaterals

Angles that meet

Angle sums in triangles and quadrilaterals

Angles in polygons

The sum of the interior angles

Regular polygons

Angles in parallel lines

Three rules, one transversal

Symmetry in two and three dimensions

Symmetry in two dimensions

ExtendedSymmetry in three dimensions

Constructions and nets

Constructing a triangle from three sides

Nets

Scale drawings and bearings

Three-figure bearings

Scale drawings

Similarity — lengths

Similar shapes

Showing two triangles are similar

Similarity — areas and volumes

Why the powers change

Circle theorems — centre and semicircle

The three foundation theorems

Circle theorems — segments

Three theorems about segments

Circle theorems — chords and tangents

ExtendedThree symmetry properties

Exam advice

Common mistakes

Model answer

Recall checklist

Every Mathematics topic, in one PDF you keep

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