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

These revision notes cover 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). The specification states that Extension papers can assess all content, so Core content is also Extension content. Two statements are Extension only: the conditions for congruent triangles (G7) and constructing plans and elevations (G11).

Explanations and fuller worked examples are in the study guide. Then test yourself with the practice questions. The course page is OxfordAQA IGCSE Mathematics and the printable list is the checklist. The free 10-minute diagnostics show which topics need work first.

Key definitions

Term	Meaning
Line of symmetry	A mirror line that splits a shape into two matching halves
Order of rotation symmetry	Number of times a shape fits onto itself in one full turn, counting the start. Order 1 means none
Congruent	Same shape and same size. Flips and turns are allowed
Similar	Same shape, different size allowed. Matching angles equal, matching lengths in one ratio
Scale factor	New length divided by matching original length
Face	A flat surface of a solid
Edge	Where two faces or surfaces meet
Vertex	A corner where edges meet
Plan	View from directly above
Elevation	View from the front or a stated side

Symmetry facts (G6)

Shape	Lines	Order
Scalene triangle	0	1
Isosceles triangle	1	1
Equilateral triangle	3	3
Parallelogram	0	2
Rectangle	2	2
Rhombus	2	2
Kite	1	1
Isosceles trapezium	1	1
Square	4	4
Regular n-gon	n	n

Rectangles, rhombuses and parallelograms (that are not squares) all have order 2. Only the rectangle and rhombus have mirror lines. A regular n-gon fits onto itself after each turn of 360°/n.

Method in steps: finding symmetry from coordinates

1. Plot or sketch the shape.
2. For rotation symmetry, find the midpoint of a pair of opposite vertices. If the shape fits onto itself after a half-turn about it, the order is at least 2.
3. For mirror lines, test lines through the centre: along a diagonal, or through midpoints of opposite sides.
4. State the equation of any line, and the coordinates of the centre.

Worked reminder

The triangle with vertices (1, 2), (7, 2) and (4, 6) has two equal sides of length 5, so it is isosceles. Its line of symmetry is x = 4 and its order of rotation symmetry is 1.

Congruence and its conditions (G7)

Congruent shapes have equal matching sides and equal matching angles. Write vertices in matching order.

Extension only: conditions for congruent triangles

Condition	What must match
SSS	Three sides
SAS	Two sides and the angle between them
ASA or AAS	Two angles and one matching side
RHS	A right angle, the hypotenuse and one other side

Not enough: AAA (same shape, size can differ) and SSA (can give two triangles).

Method in steps: a congruence argument (Extension only)

1. Name the two triangles with vertices in matching order.
2. List each equal pair on its own line, with the reason: given, common side, or a known fact.
3. Name the condition, such as SSS.
4. State the conclusion about the matching parts, for example “so angle ABC = angle ADC”.

Worked reminder

In an isosceles triangle ABC with AB = AC, M is the midpoint of BC. Triangles ABM and ACM have AB = AC, BM = CM and AM common, so they are congruent by SSS. Angle AMB = angle AMC. Together the two angles make a straight line, 180°, so each is 90°.

Similar figures (G7)

scale factor k = new length ÷ matching original length
new length = k × original length

Method in steps: missing length

1. Write which sides match, such as AB with PQ.
2. Choose a matching pair where both lengths are known and work out k.
3. Multiply or divide to find the unknown length.
4. Check: if the unknown belongs to the larger shape, the answer should be larger than its matching length.

Common configurations

Picture	Matching sides
Small triangle ADE inside ABC with DE parallel to BC	AD with AB, AE with AC, DE with BC
Two triangles meeting at a point with parallel sides (a “bow tie”)	Each side with its partner on the other triangle, found from the vertex names

Use the whole side AB, not the part DB, when AD is the small side.

Must-know distinctions

• Congruent means same size, so k = 1. Similar allows any k.
• Equal angles are enough to prove two triangles are similar, but two rectangles with equal angles need not be similar, because their sides must also be in the same ratio.
• To test rectangles, compare the long side with the long side and the short side with the short side.

Solids (G10)

Solid	Flat faces	Curved surfaces	Edges	Vertices
Cube	6	0	12	8
Cuboid	6	0	12	8
Prism with n-sided cross-section	n + 2	0	3n	2n
Pyramid with n-sided base	n + 1	0	2n	n + 1
Cylinder	2	1	2	0
Cone	1	1	1	1
Sphere	0	1	0	0

The cylinder and cone edges are curved. A prism has a constant cross-section, and its end faces are identical. A pyramid has a polygon base and one apex.

Worked reminder: a prism with 9 faces has n + 2 = 9, so n = 7. It has 14 vertices and 21 edges.

Plans and elevations (G11)

Solid	Plan	Front elevation
Cuboid	Rectangle	Rectangle
Cylinder standing on its end	Circle	Rectangle
Cone on its base	Circle and centre point	Triangle
Square-based pyramid	Square with lines to the centre	Triangle
Sphere	Circle	Circle

Method in steps: interpreting a cube model

1. Write the number of cubes in each position of the plan.
2. For the front elevation, take the largest number in each column.
3. For a side elevation, take the largest number in each row. From the right, the front row is on the left.
4. Add all the numbers for the total.

Extension only: constructing views

Draw on squared paper to a stated scale. Put the plan above or below the front elevation with matching widths in line. Put the side elevation beside the front elevation with matching heights in line. Label the dimensions.

Quick self-test

1. How many mirror lines does a regular hexagon have, and what is its order of rotation symmetry?
2. A rhombus is not a square. State its order of rotation symmetry.
3. Triangle ABC is congruent to triangle DEF, with BC = 7.2 cm. Which length in triangle DEF equals 7.2 cm?
4. (Extension only) Two triangles have sides 4 cm, 5 cm and 6 cm. State why they are congruent.
5. (Extension only) Why is AAA not enough to prove congruence?
6. A rectangle 8 cm by 5 cm is enlarged to a similar rectangle with longer side 20 cm. Find the shorter side.
7. Triangles ABC and PQR are similar with AB = 10 cm, PQ = 14 cm and BC = 15 cm. Find QR.
8. A prism has 9 faces. How many vertices does it have?
9. State the number of flat faces, curved surfaces, edges and vertices of a cone.
10. A cylinder of radius 2.5 cm and height 6 cm stands on a circular end. Describe its plan and front elevation.
11. A cube model has plan numbers: back row 4, 1 and front row 0, 3. Find the front elevation heights from left to right.
12. For the model in question 11, give the side elevation heights viewed from the right, and the total number of cubes.

Answers

1. 6 lines of symmetry and order 6.
2. Order 2.
3. EF, because BC matches EF.
4. SSS: all three pairs of sides are equal.
5. Equal angles give triangles of the same shape, but one can be an enlargement of the other.
6. k = 20/8 = 2.5, so the shorter side is 5 × 2.5 = 12.5 cm.
7. k = 14/10 = 1.4, so QR = 1.4 × 15 = 21 cm.
8. n + 2 = 9 gives n = 7, so 2n = 14 vertices.
9. 1 flat face, 1 curved surface, 1 edge, 1 vertex.
10. The plan is a circle of diameter 5 cm. The front elevation is a rectangle 5 cm wide and 6 cm high.
11. Column maxima: 4 and 3, so the heights are 4, 3.
12. Row maxima are 4 (back) and 3 (front). From the right the front is on the left, so the heights are 3, 4. The total is 4 + 1 + 0 + 3 = 8 cubes.

Where marks are usually lost

• Giving order 0 instead of order 1 for a shape with no rotation symmetry.
• Saying a parallelogram has two lines of symmetry.
• Writing a congruence statement with the vertices in the wrong order.
• Giving AAA or SSA as a congruence condition.
• Missing out the reason beside each equal pair in a congruence argument.
• Matching sides by position on the page rather than by the vertex letters.
• Using a part of a side, such as DB, in place of the whole side.
• Counting the curved surface of a cylinder as a face.
• Mixing up the left and right order of a side elevation.
• Forgetting units, or giving the dimensions without labelling a drawn view.

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.