Ever noticed how a tiny photo can look exactly like a giant poster, just bigger? That’s the magic of similarity – and it works the same way with triangles in maths.
💡 In Simple Words: Two triangles are similar when their shapes match perfectly, even if one is bigger or smaller. All their angles are the same and their sides grow in the same proportion.
What is Triangle Similarity?
Similarity means same shape, different size. Imagine tracing a tiny triangle onto a piece of paper, then enlarging it without changing any angles – you’ve got a similar triangle. The word “similar” is a fancy way of saying “looks the same after scaling”.
Key criteria for proving similarity
There are three quick ways (criteria) to show two triangles are similar. We write them as AA, SAS and SSS. Don’t worry – the letters stand for simple ideas.
- AA (Angle‑Angle): If two angles of one triangle match two angles of another, the triangles are similar. The third angles automatically match too.
- SAS (Side‑Angle‑Side): If the ratio of two sides of one triangle equals the ratio of the corresponding two sides of another triangle and the included angles (the angles between those sides) are equal, the triangles are similar.
- SSS (Side‑Side‑Side): If the three sides of one triangle are all in the same proportion as the three sides of another triangle, the triangles are similar.
Think of it like a recipe: AA is like checking two ingredients are the same, SAS adds a pinch of spice (the angle), and SSS is making sure every ingredient is scaled the same way.
Worked example 1 – AA criterion
In the diagram below, ∠A = 40° and ∠B = 70° in triangle ABC. Triangle DEF has ∠D = 40° and ∠E = 70°. Since two angles match, ABC ∼ DEF (the ~ symbol means “is similar to”).
Now, if AB = 6 cm and DE = 9 cm, the scale factor is 9/6 = 1.5. So every side of DEF is 1.5 times the corresponding side of ABC. If BC = 8 cm, then EF = 8 × 1.5 = 12 cm.
Worked example 2 – SAS criterion
Triangle GHI has sides GH = 5 cm, HI = 7 cm and included angle ∠H = 60°. Triangle JKL has sides JK = 10 cm, KL = 14 cm and the same included angle ∠K = 60°. The side ratios are 10/5 = 2 and 14/7 = 2, and the included angles are equal, so GHI ∼ JKL by SAS.
Because the scale factor is 2, any length in JKL is twice the matching length in GHI. If GI = 9 cm, then JL = 9 × 2 = 18 cm.
Worked example 3 – SSS criterion
Suppose triangle MNO has sides 4 cm, 6 cm, 8 cm. Triangle PQR has sides 8 cm, 12 cm, 16 cm. All three side ratios are 8/4 = 12/6 = 16/8 = 2. Since the ratios match, the triangles are similar by SSS. The scale factor is 2.
Quick comparison of the three criteria
| Criterion | What you need | How you check |
|---|---|---|
| AA | Two pairs of equal angles | Match the angles; the third pair follows automatically |
| SAS | Two pairs of sides in the same ratio + the included angle equal | Compute side ratios, confirm the angle between them is equal |
| SSS | All three side pairs in the same ratio | Check each side ratio; if all equal, triangles are similar |
How to use similarity in ICSE exams
ICSE questions love to hide a simple ratio behind a picture. Spot the criterion first, then write down the proportion. Once you have the scale factor, you can find missing lengths, heights or even areas (area scales by the square of the factor).
- Look for given angles – AA is the fastest.
- If a side ratio is given, see if the included angle matches – that’s SAS.
- When three side lengths are provided, compare the three ratios – that’s SSS.
- Remember: corresponding sides are always opposite the same angles.
- For area problems, use Area₁ / Area₂ = (scale factor)².
📝 Likely Exam Questions
- Question: In triangle ABC, ∠A = 30°, ∠B = 70°. Triangle DEF has ∠D = 30°, ∠E = 70°. If AB = 5 cm and DE = 8 cm, find EF when BC = 9 cm.
Answer: By AA, triangles are similar. Scale factor = DE/AB = 8/5 = 1.6. EF = BC × 1.6 = 9 × 1.6 = 14.4 cm. - Question: Triangle PQR has sides 3 cm, 4 cm, 5 cm. Triangle XYZ is similar to PQR and has a side of 9 cm corresponding to the 3 cm side. Find the perimeter of XYZ.
Answer: Scale factor = 9/3 = 3. All sides are multiplied by 3 → 9 cm, 12 cm, 15 cm. Perimeter = 9 + 12 + 15 = 36 cm. - Question: Two triangles are similar with a scale factor of ½. If the larger triangle’s area is 48 cm², what is the area of the smaller triangle?
Answer: Area scales by the square of the factor: (½)² = ¼. Smaller area = 48 × ¼ = 12 cm².