Congruence & Similarity

Revise Congruence & Similarity for International Mathematics 0607 (O Level) — revision notes and instant AI marking.

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Cambridge (CIE) IGCSE — International Maths: Extended

Congruence & Similarity

Big idea: Two shapes are congruent if they're exact copies (same shape, same size), and similar if they're the same shape but scaled up or down — and once you know the scale factor for lengths, you can work out the scale factor for areas (square it) and volumes (cube it).

Summary Everything in this chapter, at a glance

Congruent shapes are identical in shape AND size — one might be flipped, turned or slid, but never stretched.
Similar shapes have the same shape but different size — one is an enlargement of the other, and corresponding sides are all in the same ratio.
To prove two triangles are similar, you only need to show their angles match — the sides automatically fall into proportion.
To prove two triangles or shapes are congruent, you show matching side lengths and angles (not direction).
For similar shapes, the scale factor = a length on one shape ÷ the corresponding length on the other.
If the length scale factor is k, then the area scale factor is k² and the volume scale factor is k³.
Watch out for "hourglass" and "overlapping" triangle diagrams — redraw them separately if it's hard to see which sides correspond.

1. Congruence Same shape, same size — no stretching allowed

What does "congruent" actually mean?

Imagine you photocopy a triangle. The photocopy is identical to the original — same side lengths, same angles, same area. That's congruence. You're allowed to pick the copy up and rotate it, flip it over (reflect it), or slide it (translate it) — none of those change its size or shape, so it's still congruent to the original.

What you're not allowed to do is photocopy it at 150% zoom. The moment you enlarge or shrink a shape, it stops being an exact copy — it becomes a different size, so it can no longer be congruent. It might still be similar though (more on that in section 2).

💡 Quick way to think about it

Congruent = "cut it out and it fits exactly on top of the other one." Similar = "cut it out, and it fits on top only after you resize it."

How do you prove two shapes are congruent?

You need to demonstrate both of these things:

Corresponding sides are the same length — match up each side on shape 1 with its partner on shape 2 and check the lengths agree.
Corresponding angles are the same size — match up each angle and check they agree too.

Here's the detail that trips people up: you do not need the shapes to be facing the same way. A shape and its mirror image (reflection) are still congruent, even though one looks "backwards" compared to the other. Congruence only cares about size and shape — not orientation.

Rule to remember

Congruent = same shape + same size (reflections, rotations, translations of each other are fine)

In symbols: if shape A maps onto shape B using only a reflection, rotation, and/or translation (never a resize), then A ≅ B.

Exam tip: using tracing paper

If you're ever unsure whether two shapes drawn in the exam are congruent, and the diagram is drawn to scale, trace over one shape and see if it lands exactly on top of the other (you can flip and rotate your tracing paper). If it fits perfectly, they're congruent.

Practice Question 1

Practice Question 2

2. Similarity Same shape, different size — enlargements of each other

What does "similar" mean?

similar

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Also in the full note

Proving two triangles are similar
Proving two non-triangular shapes are similar
Finding the scale factor
Two valid methods for finding a missing length
The core idea
Going backwards: from area or volume to length
Step-by-step method for any missing length/area/volume question

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Congruence & Similarity