Congruence & Similarity
Revise Congruence & Similarity for International Mathematics 0607 (O Level) — revision notes and instant AI marking. Free to start.
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).
- 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.
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).
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.
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.
Look at five L-shaped tiles, A, B, C, D and E. Shape A has arms of length 4 cm and 2 cm. Shape C is a rotated version of A with identical arm lengths. Shape D is a reflected version of A with identical arm lengths. Shape B has arms of length 5 cm and 2.5 cm (an enlargement of A). Which shapes are congruent to A?
Triangle PQR has sides PQ = 6 cm, QR = 8 cm, RP = 10 cm and angle Q = 90°. Triangle XYZ has sides XY = 8 cm, YZ = 6 cm, ZX = 10 cm and angle Y = 90°. Are the two triangles congruent? Explain your reasoning.
What does "similar" mean?
Two shapes are similar if they have exactly the same shape, but possibly a different size. That means:
- All corresponding angles are equal
- All corresponding sides are in the same ratio (proportion) — i.e. one shape is a scaled version of the other
Think of a passport photo and a poster-sized print of the same photo. They look identical in shape — same proportions, same angles — but one is much bigger. That's similarity.
Similarity does NOT imply congruence. Two shapes can be similar to each other without being the same size — in fact they usually aren't. And two similar shapes could each be a different enlargement of some common "parent" shape (e.g. one is ×2 and the other is ×3 of a third shape) — they don't have to relate to each other in a simple way, just proportionally.
Proving two triangles are similar
This is the single most useful skill in this chapter, because it comes up constantly in exams. Here's the key fact: for triangles specifically, you only need to prove the angles match. You do not need to check the side ratios separately — if all three angles in one triangle equal the three angles in the other, the sides are automatically in proportion. This is a huge shortcut compared to proving general shapes are similar.
To spot equal angles, look out for these classic angle facts:
- Isosceles triangles — the two base angles are equal
- Vertically opposite angles — formed when two lines cross, equal in pairs
- Angles on parallel lines — alternate angles and corresponding angles are equal when a line crosses a pair of parallel lines
When the exam asks you to prove two triangles are similar, you must lay it out formally: for each pair of matching angles, state that they're equal, and give the specific geometric reason (e.g. "vertically opposite angles are equal"). A vague "they look the same" earns zero marks — examiners want named angles and named reasons.
A pair of similar triangles very often appears "opposite" each other in exactly this hourglass shape inside a bigger diagram. The moment you see two lines crossing with parallel lines involved, hunt for the vertically opposite angle at the crossing point — it's almost always one of your three equal angles.
Proving two non-triangular shapes are similar
Rectangles, quadrilaterals, or any other shape with more than 3 sides don't get the "angles are enough" shortcut — equal angles alone don't guarantee proportional sides for a 4+ sided shape (think of a square vs. a non-square rectangle: same angles, different side ratios!). So for these, you must show that every pair of corresponding sides has the same scale factor.
Worked example
Two rectangles: Rectangle 1 is 15 cm × 5 cm. Rectangle 2 is 6 cm × 2 cm.
If a diagram has similar triangles tangled up or overlapping (like the hourglass above), it's often much easier to redraw them separately, facing the same direction, before you start matching up sides or angles. Fewer mistakes happen when you're not squinting at a crossed-over diagram.
In a diagram, AB and CD are parallel straight lines that cross at point X (an hourglass shape, as shown above, with A and B on one side, C and D on the other). Show that triangle ABX is similar to triangle CDX.
Quadrilateral 1 has sides 8 cm, 12 cm, 10 cm, and 6 cm (in order). Quadrilateral 2 has corresponding sides 4 cm, 6 cm, 5 cm, and 3.5 cm. Is Quadrilateral 2 similar to Quadrilateral 1? Show your working.
Finding the scale factor
Once you know two shapes are similar, the scale factor tells you exactly how much bigger (or smaller) one is than the other.
- If shape 2 is bigger than shape 1 → the scale factor from 1 to 2 is greater than 1.
- If shape 2 is smaller than shape 1 → the scale factor from 1 to 2 is between 0 and 1 (a fraction).
Two valid methods for finding a missing length
Both of these get you to the same correct answer — pick whichever feels more natural to you.
Method 1 — scale factor from shape 1 → shape 2 (can be a fraction)
- Step 1: Divide a known length on shape 2 by the corresponding length on shape 1. (This scale factor can be less than 1.)
- Step 2: To find a missing length on shape 2, multiply the corresponding length on shape 1 by the scale factor. To find a missing length on shape 1, divide the corresponding length on shape 2 by the scale factor.
Method 2 — scale factor from the smaller shape → the bigger shape (always > 1)
- Step 1: Divide a known length on the bigger shape by the corresponding length on the smaller shape. (This scale factor is always greater than 1 — easier to keep track of mentally.)
- Step 2: To find a missing length on the bigger shape, multiply by the scale factor. To find a missing length on the smaller shape, divide by the scale factor.
Method 2 is often less error-prone because the scale factor is always a "nice" number bigger than 1 — no messing around with awkward fractions like 1/2 or 2/3. But some students find Method 1 more intuitive since it directly matches shape 1 → shape 2 without needing to figure out which shape is "smaller" first. Use whichever one clicks for you — just be consistent so you don't mix them up mid-question.
Worked example
ABCD and PQRS are similar shapes. AB = 6 cm corresponds to PQ = 3 cm. AD = 15 cm. Find the length of PS (which corresponds to AD).
PS = AD × 1/2 = 15 × 1/2 = 7.5 cm
PS (on the smaller shape) = AD ÷ 2 = 15 ÷ 2 = 7.5 cm
Both methods agree: PS = 7.5 cm. That agreement is a great way to self-check your answer in an exam if you have time.
Students often multiply when they should divide (or vice versa) because they lose track of which shape is bigger. Before you calculate anything, quickly ask: "Is the length I'm looking for on the bigger shape or the smaller shape?" That answer tells you whether to multiply or divide.
Triangle 1 has a base of 4 cm. Triangle 2 is similar to Triangle 1 with a corresponding base of 10 cm. If Triangle 1's height is 3.2 cm, find Triangle 2's corresponding height.
Two similar cones have corresponding slant heights of 24 cm and 9 cm. The larger cone has a base radius of 16 cm. Find the base radius of the smaller cone.
The core idea
This is where a lot of students trip up, so let's slow right down. If you enlarge a shape by a length scale factor of, say, 2 — meaning every length doubles — you might assume the area also just doubles. It doesn't. Because area is a two-dimensional measurement (length × length), it gets scaled by the length factor twice. So area scales by 2² = 4, not 2. Volume is three-dimensional (length × length × length), so it scales by 2³ = 8.
See the pattern? The length scale factor is k. The area scale factor is k². The volume scale factor is k³. This relationship holds for any pair of similar shapes or solids — spheres, cylinders, pyramids, irregular blobs, anything — as long as they really are mathematically similar (same shape, uniformly scaled).
Area scale factor = k²
Volume scale factor = k³
Going backwards: from area or volume to length
Sometimes the exam gives you area or volume information and wants a length — so you need to reverse the process using roots instead of powers.
Volume scale factor = (√(area scale factor))³
Area scale factor = (∛(volume scale factor))²
Step-by-step method for any missing length/area/volume question
- Step 1: Identify the two known, equivalent quantities you're given (are they lengths, areas, or volumes?).
- Step 2: Find that specific scale factor: scale factor = (second quantity) ÷ (first quantity).
- Step 3: Convert to whichever scale factor you actually need (length k, area k², or volume k³) using the square/cube or square-root/cube-root relationships above.
- Step 4: Multiply or divide by that scale factor to get your missing quantity — and sanity-check: should your answer be bigger or smaller than the value you started with?
Worked example
Solid A and solid B are mathematically similar. Volume of A = 32 cm³. Volume of B = 108 cm³. Height of A = 10 cm. Find the height of B.
Volume B (108) is bigger than volume A (32), so B is the larger solid — meaning its height should also be bigger than A's height (10 cm). We got 15 cm, which is indeed bigger. If you'd got a height smaller than 10 cm here, that would be your signal something went wrong (probably multiplied instead of divided, or vice versa).
Two similar cylinders have radii 4 cm and 10 cm. The smaller cylinder has a curved surface area of 60 cm². Find the curved surface area of the larger cylinder.
Two similar statues are made from the same material. The smaller statue has a mass of 5 kg and a height of 20 cm. The larger statue has a mass of 135 kg. Find the height of the larger statue. (Hint: mass is proportional to volume for objects of the same material/density.)
- Method 2 — scale factor from the smaller shape → the bigger shape (always > 1)
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