6 Geometry

Learning objectives

6A Euclidean geometryGeometrical properties of Euclidean space, as listed below In solving any problem or rider, students may use any knowledge they possess Solutions may be by traditional methods (e.g. congruent triangles), vectors, the use of transformations such as translation, reflection, rotation and enlargement, or a mixture of these Formal proofs of theorems will not be required 6B Geometrical reasoningGeometrical reasoning 6C Angle propertiesAngle properties of parallel lines, triangles and polygons, including regular polygons Angles on a straight line, angles around a point Angles measured anticlockwise will be taken as positive; clockwise as negative 6D QuadrilateralsProperties of the parallelogram, rectangle, square, rhombus, trapezium and kite 6E SymmetrySymmetry about a point, line or plane Recognise line and rotational symmetry Complete shapes with a given axis of symmetry and order of rotational symmetry 6F Pythagoras in 2D and 3DUse of Pythagoras’ theorem in 2D and 3D Including its use in any acute–angled triangle where an altitude is given or constructed The angle bisector theorems are excluded 6G SimilaritySimilarity: areas and volumes of similar figures Understanding how scale factors are related to area and volume 6H Triangle similarityProve the similarity of two triangles 6I Congruent shapesCongruent shapes 6J Triangle congruenceUnderstand and use SSS, SAS, ASA and RHS conditions to prove the congruence of triangles 6K Circle theoremsChord, angle and tangent properties of circles To include knowledge of the intersecting chord properties (both internal and external) and the alternate segment theorem 6L Cyclic quadrilateralsProperties of a cyclic quadrilateral 6M LociLoci in two dimensions ‘Tracing paper’ methods will not be acceptable 6N ConstructionsConstructions of bisector of an angle and of perpendicular bisector (mediator) of a straight line Constructions using only ruler and compasses

A geometry problem is a chain from facts you are given to a fact you need. Mark the diagram, name each usable relationship, and choose the shortest chain that reaches the target.

Route	Useful when
angle facts	the target is an angle or parallel relationship
congruence or similarity	corresponding lengths or angles must be transferred
transformations	one shape is moved, reflected, rotated or enlarged
vectors	parallelism, ratios or a common point can be expressed algebraically

Start with statements that are guaranteed by labels or the question, not by appearance. Add one justified consequence at a time; if a route stalls, return to the givens and try a different valid route.

Formal theorem proofs are not required, but every step in a solution still needs a valid geometric reason. A diagram that is not accurately drawn is never evidence that lines are equal, parallel or perpendicular.

Write geometry as paired statements: make a claim, then give the fact that guarantees it. This exposes missing assumptions and makes a multi-step argument checkable.

Claim	Suitable reason
$ngle ABC=ngle BCD$	alternate angles, because $AB\parallel CD$
$OA=OB$	radii of the same circle
$riangle ABC\sim riangle DEF$	two corresponding angles are equal
$AC=DF$	corresponding sides of congruent triangles

Name points in matching order, state any intermediate equality you need, and finish by explicitly connecting the established result to the requested conclusion.

Do not write a theorem name without showing that its conditions hold. For example, alternate angles are equal only after the relevant lines have been established as parallel.

Structure	Angle fact
straight line	sum $180^\circ$
around a point	sum $360^\circ$
triangle	sum $180^\circ$
quadrilateral	sum $360^\circ$
interior angles of an $n$ -gon	sum $(n-2)180^\circ$
regular $n$ -gon exterior angle	$360^\circ/n$

With parallel lines, corresponding and alternate angles are equal; co-interior angles sum to $180^\circ$ . Mark the two parallel lines and the transversal before naming the relationship.

\text{anticlockwise turn}>0,\qquad \text{clockwise turn}<0

Use only marked parallel lines or established facts. Do not transfer an angle merely because two lines look parallel, and do not confuse a polygon's interior-angle sum with one interior angle of a regular polygon.

Shape	Guaranteed properties
parallelogram	opposite sides parallel and equal; opposite angles equal; diagonals bisect each other
rectangle	parallelogram with four right angles; diagonals equal
rhombus	parallelogram with four equal sides; diagonals perpendicular
square	rectangle and rhombus properties
trapezium	one pair of opposite sides parallel
kite	two pairs of adjacent equal sides; one diagonal bisects the other at right angles

Classify from the properties that are guaranteed, not from appearance. A square is also a rectangle, rhombus and parallelogram because it satisfies all their defining conditions.

When solving, translate each mark into a property: arrows mean parallel, ticks mean equal lengths, a small square means a right angle. Then select only conclusions licensed by those marks.

Diagonals are not automatically equal, perpendicular or angle bisectors in every quadrilateral. State the specific shape property before using one of these conclusions.

A line of symmetry reflects a shape onto itself. Corresponding points lie on opposite sides of the mirror line, at equal perpendicular distances from it.

Rotational symmetry of order $n$ means the shape matches itself $n$ times in one full turn. The smallest positive matching angle is $360^\circ/n$ ; order 1 means only the full turn works.

Symmetry	Fixed object
about a point	the centre of rotation
about a line	every point on the mirror line
about a plane	every point in the mirror plane

To complete a reflected shape, count perpendicular grid steps from the axis rather than copying a horizontal or vertical offset. Do not count the starting position twice when finding rotational order.

a^2+b^2=c^2\qquad(c\text{ is opposite the right angle})

Find or create a right-angled triangle, label the hypotenuse first, substitute lengths, then take the positive square root. In an acute triangle, an altitude splits the shape into right-angled triangles.

\text{cuboid space diagonal}=\sqrt{l^2+w^2+h^2}

Pythagoras applies only to a right-angled triangle. In 3D, use a visible face diagonal as an intermediate length if necessary; never add unsquared lengths, and do not use the excluded angle-bisector theorems.

\text{length factor}=k,\qquad \text{area factor}=k^2,\qquad \text{volume factor}=k^3

Write the direction of comparison before forming a ratio. If shape B is an enlargement of A by $k$ , every corresponding length in B is $k$ times its partner in A.

k=\sqrt{\frac{A_B}{A_A}}\quad\text{or}\quad k=\sqrt[3]{\frac{V_B}{V_A}}

Square or cube the linear factor, not the measured area or volume. Similar figures have equal corresponding angles and proportional corresponding lengths; equal area alone does not prove similarity.

To prove similarity, establish enough corresponding information: AA uses two equal angle pairs; SSS uses three proportional side pairs; SAS uses two proportional side pairs with the included angle equal.

\triangle ABC\sim\triangle DEF\ \Rightarrow\ A\leftrightarrow D,\ B\leftrightarrow E,\ C\leftrightarrow F

After proving similarity, use the same correspondence order to equate angle pairs or form side ratios. Keep one consistent scale-factor direction throughout the calculation.

Equal angles prove the same shape, not the same size. Do not assume similarity because triangles look alike, and do not mix non-corresponding sides in a proportion.

Congruent shapes have exactly the same size and shape. One can be translated, rotated or reflected to fit the other without enlargement.

Match distinctive vertices, equal angles and equal sides, then list vertices in corresponding order. Orientation may reverse after a reflection, but corresponding lengths remain equal.

Relationship	Angles	Corresponding lengths
congruent	equal	equal
similar	equal	proportional

Equal area, equal perimeter or the same general outline alone does not guarantee congruence. A scale factor other than 1 gives similar, not congruent, shapes.

Test	Information required
SSS	all three corresponding sides equal
SAS	two sides and their included angle equal
ASA	two angles and the included side equal
RHS	right triangles with equal hypotenuse and one other side

State the three matching facts with reasons, name the test, then conclude using correctly ordered triangle names. Only after congruence is established may you transfer other corresponding sides or angles.

\triangle ABC\cong\triangle DEF\ \Rightarrow\ AB=DE,\ BC=EF,\ AC=DF

AAA proves similarity, not congruence. SSA is not a general congruence test, and in SAS the known angle must be between the two known sides.

Configuration	Theorem
same chord or arc	angles at the circumference are equal
same arc	angle at centre = twice angle at circumference
angle in a semicircle	$90^\circ$
radius and tangent at contact	perpendicular
tangent and chord	angle equals angle in alternate segment

Intersections	Product relationship
chords intersect inside at $P$	$PA\cdot PB=PC\cdot PD$
two secants from external $P$	$PA\cdot PB=PC\cdot PD$ using whole secants
tangent $PT$ and secant $PAB$	$PT^2=PA\cdot PB$

Identify the chord, arc, centre, tangent or intersection that triggers a theorem. Mark equal radii and add triangle angle facts only after the circle relationship is clear.

A tangent theorem needs the exact point of contact. For an external secant product, multiply each external segment by its whole secant, not by the internal segment alone.

A cyclic quadrilateral has all four vertices on one circle. Its opposite interior angles are supplementary.

\angle A+\angle C=180^\circ,\qquad \angle B+\angle D=180^\circ

An exterior angle of a cyclic quadrilateral equals the interior opposite angle, because the adjacent interior angle and the exterior angle also sum to $180^\circ$ .

The sides need not be equal and the diagonals need not be diameters. Do not use the supplementary-opposite-angle rule unless all four vertices are established on the same circle.

Condition	Locus
fixed distance $r$ from point $A$	circle centre $A$ , radius $r$
equidistant from points $A$ and $B$	perpendicular bisector of $AB$
fixed perpendicular distance from a line	two parallel lines
equidistant from two intersecting lines	both angle bisectors

For 'less than' or 'within', shade the side or interior satisfying the inequality; for 'more than', use the opposite region. Intersections of conditions give the feasible set.

Test one point from each candidate region against the original distance wording. Keep construction arcs and boundaries visible when an exact locus is required.

A locus is the complete set of points satisfying a condition, not one example path. Use geometric construction or measured distance; tracing-paper methods are not acceptable.

Angle bisector: from the vertex draw an arc cutting both arms; from those two cut points draw equal-radius arcs that intersect; join the vertex to that intersection.

Perpendicular bisector of segment $AB$ : with compass radius greater than half $AB$ , draw arcs from $A$ and $B$ above and below the segment; join the two arc intersections.

Construction	Guaranteed result
angle bisector	points on it are equidistant from the two arms
perpendicular bisector	points on it are equidistant from $A$ and $B$ ; it crosses $AB$ at $90^\circ$ and its midpoint

Use only an unmarked ruler and compasses, preserve the construction arcs, and keep the same compass radius for paired arcs. Measuring halfway with ruler marks or estimating an angle is not a valid construction.

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6 Geometry

Learning objectives

Solve geometry by linking known facts

Build a convincing geometric argument

Chase angles systematically

Identify quadrilaterals from defining properties

Recognise and complete symmetry

Use Pythagoras in 2D and 3D

Connect length, area and volume scale factors

Prove two triangles are similar

Decide when shapes are congruent

Prove triangle congruence with four tests

Choose and apply circle theorems

Use cyclic quadrilateral properties

Translate distance conditions into loci

Construct angle and perpendicular bisectors

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