OxfordAQA International GCSE Mathematics: Symmetry, congruence, similarity and 3D solids (9260)

This study guide covers references G6, G7, G10 and G11 in section 3.3.1 of the OxfordAQA International GCSE Mathematics (9260) specification, Version 5.1 (for exams May/June 2018 onwards). It teaches symmetry of 2D shapes, congruence and similarity, the parts of 3D solids, and plans and elevations. The specification states that Extension papers can assess all content, so Core content is also Extension content. Only the extra Extension statements carry the label “Extension only”.

Pair this guide with the revision notes and the practice questions. The course page is OxfordAQA IGCSE Mathematics and the printable list is the checklist. To find weak spots first, try a free 10-minute diagnostic.

What this guide covers

Reference	Skill	Level
G6	Recognise reflection and rotation symmetry of 2D shapes	Core
G7	Understand congruence and similarity	Core
G7	Calculate lengths of similar figures	Core
G7	Understand and use conditions for congruent triangles	Extension only
G10	Identify properties of faces, surfaces, edges and vertices of the standard solids (cube, cuboid, prism, cylinder, pyramid, cone, sphere)	Core
G11	Interpret plans and elevations of 3D shapes	Core
G11	Construct plans and elevations of 3D shapes	Extension only

Angle facts, circle theorems and scale drawings are separate topics. Only lengths of similar figures are taught here.

Reflection and rotation symmetry (G6)

A 2D shape has reflection symmetry if a mirror line splits it into two halves that match exactly. The mirror line is a line of symmetry. Count the lines.

A shape has rotation symmetry if it fits onto its own outline after a turn of less than 360° about a point, the centre of rotation. The order of rotation symmetry is the number of times the shape fits onto itself during one full turn, counting the starting position. Order 1 means no rotation symmetry.

Shape	Lines of symmetry	Order of rotation symmetry
Scalene triangle	0	1
Isosceles triangle	1	1
Equilateral triangle	3	3
Parallelogram (not a rectangle or rhombus)	0	2
Rectangle with unequal sides	2	2
Rhombus with no right angles	2	2
Kite	1	1
Isosceles trapezium	1	1
Square	4	4
Regular polygon with n sides	n	n
Circle	infinitely many	infinite

For a regular polygon with n sides, the smallest turn that fits it onto itself is 360°/n. A regular decagon turns through 36° and has order 10.

Worked example 1: a shape on a grid

A quadrilateral has vertices A(0, 0), B(4, 0), C(6, 3) and D(2, 3). State its symmetry.

AB and DC both run 4 across. AD and BC both go 2 right and 3 up.
Opposite sides are equal and parallel, so the shape is a parallelogram.
Midpoint of AC = (3, 1.5), which is also the midpoint of BD.
A half-turn about (3, 1.5) sends A to C and B to D.
The sides AB and AD are different lengths and the angles at A and B differ,
so no mirror line maps the shape onto itself.

Answer: no lines of symmetry, and rotation symmetry of order 2 about (3, 1.5).

Worked example 2: completing a design

Points (1, 1), (2, 4) and (3, 2) are part of a design with the line x = 4 as a line of symmetry. Each point has a partner the same distance from the line on the other side, so x becomes 8 − x and y stays the same. The partners are (7, 1), (6, 4) and (5, 2).

Congruence (G7, Core)

Congruent shapes match exactly in shape and size. Turning, sliding or flipping one copy can place it over the other, and a flipped copy still counts as congruent.

In congruent shapes, corresponding sides are equal and corresponding angles are equal. Write the vertices in matching order: “triangle PQR is congruent to triangle XYZ” means P matches X, Q matches Y and R matches Z.

Worked example 3: reading off matching parts

Triangle PQR is congruent to triangle XYZ. PQ = 8.4 cm, QR = 6.5 cm, angle P = 37° and angle Q = 82°. Find XY, YZ and angle Z.

P matches X, Q matches Y, R matches Z.
XY matches PQ, so XY = 8.4 cm.
YZ matches QR, so YZ = 6.5 cm.
Angle R = 180° − 37° − 82° = 61°, and Z matches R, so angle Z = 61°.

Conditions for congruent triangles (G7, Extension only)

You do not need to check all six measurements. Two triangles are congruent if any one of these holds.

Condition	Meaning
SSS	All three pairs of sides are equal
SAS	Two pairs of sides are equal and the angles between them are equal
ASA or AAS	Two pairs of angles are equal and one pair of corresponding sides is equal
RHS	Both triangles have a right angle, the hypotenuses are equal and one other pair of sides is equal

Two conditions do not work. AAA gives triangles of the same shape but possibly different sizes. SSA, meaning two sides plus an angle that is not between them, can fit two different triangles.

Worked example 4: choosing the condition

Decide whether each pair of triangles is congruent.

(i)   Triangle 1: sides 5 cm, 7 cm, 9 cm. Triangle 2: sides 9 cm, 5 cm, 7 cm.
      Three pairs of equal sides: congruent (SSS).
(ii)  Both triangles have angles 40°, 60° and 80°.
      Only angles are known: not necessarily congruent (AAA).
(iii) Both have sides 6 cm and 10 cm with a 50° angle between them.
      Congruent (SAS).
(iv)  Both have angles 35° and 75°, and an 8 cm side opposite the 35° angle.
      Congruent (AAS).

Why SSA fails

Draw a 30° angle, mark a side of 8 cm along one arm, then swing an arc of radius 6 cm from the end of that side towards the other arm. The arc can cut the other arm in two places, giving two different triangles with the same SSA information. RHS avoids this because the known angle is 90°, and the arc then touches the arm in only one place.

Worked example 5: a short congruence argument

ABCD is a kite with AB = AD and CB = CD. Show that angle ABC = angle ADC.

In triangles ABC and ADC:
  AB = AD   (given)
  CB = CD   (given)
  AC is common to both triangles.
So the triangles are congruent by SSS.
Corresponding angles in congruent triangles are equal, and angle ABC
corresponds to angle ADC, so angle ABC = angle ADC.

Similar figures and lengths (G7, Core)

Two shapes are similar if one is an enlargement of the other, possibly turned or flipped. Matching angles are equal, and every pair of matching lengths has the same ratio. This ratio is the scale factor.

scale factor k = (length in new shape) / (matching length in original shape)

To find a missing length, find k from a pair of matching sides, then multiply by k. Congruent shapes are the special case k = 1.

To test for similarity, check that every pair of matching sides gives the same ratio.

Worked example 6: similar triangles

Triangle ABC is similar to triangle DEF. AB = 9 cm, BC = 15 cm, AC = 12 cm and DE = 6 cm, where DE matches AB. Find EF and DF.

k = DE / AB = 6/9 = 2/3
EF = 2/3 × 15 = 10 cm
DF = 2/3 × 12 = 8 cm

Worked example 7: a rectangular photograph

A photo 18 cm by 12 cm is enlarged to a similar rectangle with shorter side 20 cm. Find the longer side.

k = 20 / 12 = 5/3
Longer side = 5/3 × 18 = 30 cm

Worked example 8: a line parallel to the base

In triangle ABC, point D is on AB and point E is on AC, and DE is parallel to BC. AD = 6 cm, DB = 4 cm and DE = 9 cm. Find BC.

Triangle ADE is similar to triangle ABC. The matching sides are AD with AB, and DE with BC.

AB = 6 + 4 = 10 cm
k = AB / AD = 10/6 = 5/3
BC = 5/3 × 9 = 15 cm

Do not use DB, the 4 cm part, as if it matched DE. Compare the whole side AB with the whole side AD.

Faces, surfaces, edges and vertices (G10)

A face is a flat surface. An edge is where two faces or surfaces meet. A vertex (plural vertices) is a corner where edges meet. Describe cylinders, cones and spheres using flat faces and curved surfaces.

Solid	Flat faces	Curved surfaces	Edges	Vertices
Cube	6 squares	0	12	8
Cuboid	6 rectangles	0	12	8
Triangular prism	5	0	9	6
Cylinder	2 circles	1	2 curved	0
Square-based pyramid	5	0	8	5
Cone	1 circle	1	1 curved	1
Sphere	0	1	0	0

A prism has the same cross-section all along its length: two identical parallel end faces and rectangles between them. A pyramid has a polygon as its base and triangular faces meeting at one vertex, the apex.

For a prism whose cross-section has n sides:

Faces = n + 2     Edges = 3n     Vertices = 2n

For a pyramid whose base has n sides:

Faces = n + 1     Edges = 2n     Vertices = n + 1

Worked example 9: finding the solid

A prism has 12 vertices. Find its number of faces and edges.

Vertices = 2n = 12, so n = 6 (a hexagonal prism).
Faces = 6 + 2 = 8.
Edges = 3 × 6 = 18.

Plans and elevations (G11)

A plan is the view from directly above. A front elevation is the view from the front. A side elevation is the view from a stated side. Look straight at the solid. Edges you can see are drawn as lines.

Solid (standing as described)	Plan	Front elevation
Cube or cuboid	Rectangle	Rectangle
Cylinder on a circular end	Circle	Rectangle
Cone on its base	Circle with a centre point	Triangle
Square-based pyramid	Square with lines from corners to the centre	Triangle
Sphere	Circle	Circle

Worked example 10: a triangular prism (interpret, Core)

A triangular prism is 6 cm long. Its cross-section is an isosceles triangle with base 4 cm and height 3 cm. It rests on a rectangular face with its triangular ends facing left and right.

Plan (from above)          Front elevation          Side elevation (from right)
+--------------+           +--------------+              /\
|              |  ridge    |              |             /  \  3 cm
+--------------+  line     |              |  3 cm      /    \
|              |           +--------------+            +------+
+--------------+                6 cm                      4 cm
 6 cm by 4 cm

The plan is a 6 cm by 4 cm rectangle with a ridge line 2 cm from each long edge. The front elevation is a 6 cm by 3 cm rectangle. The side elevation is the triangle.

Worked example 11: a model made of cubes

Each number is the count of cubes stacked at that position of the plan, and the back row is at the top.

Back:    3  0  1
Middle:  0  1  1
Front:   2  2  0

The front elevation shows the tallest stack in each column: 3, 2, 1 from left to right. The side elevation from the right shows the tallest stack in each row. From the right, the front is on your left, so reading left to right gives 2, 1, 3. The total is 3 + 1 + 1 + 1 + 2 + 2 = 10 cubes.

Constructing plans and elevations (Extension only)

1. Choose a scale and draw on squared paper.
2. Draw the plan, using the length and width.
3. Draw the front elevation above or below it, so that matching widths line up.
4. Draw the side elevation beside the front elevation, so that heights line up.
5. Label the dimensions and viewing direction.

Common errors

• Counting a parallelogram as having line symmetry because the diagonals look like mirror lines.
• Writing order 0 for no rotation symmetry. The order is 1.
• Using AAA or SSA as a reason for congruence.
• Using the part DB instead of the whole side AB in a parallel-line triangle.
• Calling the curved side of a cylinder a face.
• Drawing the side elevation from the wrong side, which reverses the order.

Next steps

Revise with the revision notes, then try the practice questions.

Official syllabus

OxfordAQA International GCSE Mathematics (9260), Version 5.1, for exams May/June 2018 onwards, published by OxfordAQA. Section 3.3.1, references G6, G7, G10 and G11.