Congruence, Similarity & Geometrical ProofEdexcel IGCSE Maths: Revision notes

Section 1

What makes two shapes congruent?

Two shapes are congruent if they are exactly the same shape and size — one can be mapped onto the other by a rotation, reflection, translation, or a combination of these (no resizing allowed).

For triangles, you don't need to check all three sides and all three angles match. Instead, you can prove congruence using one of four condition sets:

SSS (Side-Side-Side): all three sides are equal
SAS (Side-Angle-Side): two sides and the included angle (the angle between them) are equal
ASA (Angle-Side-Angle): two angles and the included side are equal (also covers AAS — two angles and any side)
RHS (Right angle-Hypotenuse-Side): a right angle, the hypotenuse, and one other side are equal — only valid for right-angled triangles

If one of these four conditions holds, the triangles are congruent — every other pair of corresponding sides/angles must also match.

Key termsCongruentSSSSASASARHSIncluded angle

Common mistake

SSA (two sides and a non-included angle) is NOT a valid congruence condition — it can produce two different triangles (the 'ambiguous case'). Never use it in a proof.

Exam tip

When writing a congruence proof, always state which condition you're using at the end, e.g. 'therefore triangles ABC and DEF are congruent (SAS)'.

Section 2

How do I structure a congruence proof?

A full-marks congruence proof follows a clear pattern:

State what you're proving — e.g. "Prove that triangle $ABC$ is congruent to triangle $DEF$ ."
List three matching facts, one per line, each with a reason (given information, a shared/common side, vertically opposite angles, parallel line angle rules, circle theorems, etc.)
Name the condition (SSS, SAS, ASA or RHS) that those three facts satisfy.
State the conclusion clearly, matching vertices in the correct order (e.g. $ABC \equiv DEF$ means $A \leftrightarrow D$ , $B \leftrightarrow E$ , $C \leftrightarrow F$ ).

Common sources of matching facts:

A shared/common side (appears in both triangles) is always equal to itself
Vertically opposite angles are equal
Alternate or corresponding angles on parallel lines are equal
Radii of the same circle are equal
Information given in the question (marked on a diagram or stated in words)

Key termsCommon sideVertically opposite anglesCorresponding vertices

Example

Given: $AB = AC$ and $AD$ bisects angle $A$ . Prove triangles $ABD$ and $ACD$ are congruent. $AB = AC$ (given), angle $BAD$ = angle $CAD$ (given, bisected), $AD = AD$ (common side). This is SAS, so triangle $ABD \equiv$ triangle $ACD$ .

Common mistake

Losing marks by writing the vertices in the wrong order, e.g. writing $ABC \equiv EDF$ when the correct correspondence is $ABC \equiv DEF$ . Always match up equal angles/sides carefully.

Section 3

What does it mean for shapes to be similar?

Two shapes are similar if they have the same shape but not necessarily the same size — one is an enlargement (scale factor) of the other. For similar shapes:

All corresponding angles are equal
All corresponding sides are in the same ratio (the scale factor)

For triangles, similarity can be shown using AA (two pairs of equal angles — the third pair must then also be equal, since angles in a triangle sum to $180°$ ), or by showing all corresponding sides are in the same ratio (SSS similarity), or two sides in the same ratio with the included angle equal (SAS similarity).

The scale factor for area is the square of the linear scale factor, and the scale factor for volume is the cube of the linear scale factor. If the linear scale factor is $k$ , area scale factor is $k^2$ and volume scale factor is $k^3$ .

Key termsSimilarScale factorAA similarityLinear scale factor

Think of it like this

Think of similar shapes like photocopies at different zoom settings — the picture looks identical, just bigger or smaller. Congruent shapes are photocopies at exactly 100% zoom.

Exam tip

To find a missing side in similar shapes, find the scale factor first using a pair of known corresponding sides, then multiply (or divide) to find the unknown.

Section 4

How do I solve similar shape problems involving area or volume?

Exam questions often give an area or volume ratio and ask for a length, or vice versa. Work systematically:

Identify whether the given ratio is for length, area, or volume.
Convert to the linear scale factor $k$ : if area ratio is given, take the square root; if volume ratio is given, take the cube root.
Apply $k$ (or $k^2$ / $k^3$ ) to answer what's being asked.

For example, if two similar cylinders have volumes in ratio $8:27$ , the linear scale factor is $\sqrt[3]{27/8} = 3/2$ . If the smaller cylinder's height is $4\text{cm}$ , the larger's height is $4 \times 3/2 = 6\text{cm}$ .

Always double check whether the question is asking you to scale up or down, and whether you need length, area, or volume as your final answer — these are the most common places marks are lost.

Key termsArea scale factorVolume scale factor

Common mistake

Forgetting to take the square/cube root when converting from an area or volume ratio back to a linear scale factor — this is the single most common error in this topic.

Example

Two similar triangles have areas $18\text{cm}^2$ and $50\text{cm}^2$ . Area ratio $= 18:50 = 9:25$ , so linear scale factor $= \sqrt{9/25} = 3/5$ .

Must Know

The four congruence conditions are SSS, SAS, ASA (and AAS), and RHS — SSA is never valid
A congruence proof needs three justified matching facts, the named condition, and vertices in matching order
Similar shapes have equal corresponding angles and sides in the same ratio; AA is enough to prove triangle similarity
Linear scale factor $k$ gives area scale factor $k^2$ and volume scale factor $k^3$
Common sides, vertically opposite angles, and parallel line angle facts are the usual 'free' reasons in proofs
Always write the conclusion with corresponding vertices in the correct order (e.g. $ABC \equiv DEF$ )

That's the notes covered.

Carry on to the next subtopic.