Ever wondered why a tiny triangle on a map can tell you the real distance between two cities?
In simple words, two triangles are similar when they have the same shape but maybe different sizes. All their angles match, and their sides grow or shrink by the same factor.
What does "similar triangles" really mean?
When we say two triangles are similar, we mean you could place one on top of the other after scaling it up or down, and every corner would line up perfectly. Think of it like a photo you zoom in or out – the picture stays the same, only the size changes.
Key terms explained
- Corresponding angles: angles that sit in the same relative position in each triangle. If you rotate one triangle to match the other, these angles sit on top of each other.
- Corresponding sides: the sides that sit opposite the corresponding angles. They are the “matching” sides.
- Scale factor: the number you multiply each side of the smaller triangle by to get the side of the bigger one. It’s like the recipe multiplier when you double a cake recipe.
How to prove triangles are similar – the three theorems
ICSE geometry gives you three reliable shortcuts, called similarity criteria. If any one of them fits, the triangles are definitely similar.
1. AA (Angle‑Angle) criterion
If two angles of one triangle are equal to two angles of another triangle, the third angles must also be equal (because the angles in a triangle always add up to 180°). So the triangles are similar.
2. SAS (Side‑Angle‑Side) criterion
Here you need one pair of equal angles and the sides that form those angles must be in the same proportion. For example, if the ratio of the two sides around the angle in triangle A is the same as the ratio of the two sides around the equal angle in triangle B, the triangles are similar.
3. SSS (Side‑Side‑Side) criterion
If the three sides of one triangle are all in the same proportion as the three sides of another triangle, the triangles are similar – you don’t even need to look at the angles.
Worked example – using the AA criterion
Suppose we have triangle ABC with ∠A = 40°, ∠B = 70°, and triangle DEF with ∠D = 40°, ∠E = 70°. The remaining angles are automatically equal because each triangle’s angles sum to 180°.
Since two angles match, AA tells us the triangles are similar. The scale factor k is found by comparing any pair of corresponding sides. If AB = 6 cm and DE = 9 cm, then k = 9/6 = 1.5. Every side in triangle DEF is 1.5 times the matching side in triangle ABC.
Worked example – using the SAS criterion
Triangle PQR has sides PQ = 8 cm, QR = 6 cm and angle Q = 60°. Triangle XYZ has sides XY = 12 cm, YZ = 9 cm and angle Y = 60°. The angles are equal, and the side ratios PQ/QR = 8/6 = 4/3 and XY/YZ = 12/9 = 4/3 are the same. Because an equal angle is flanked by sides in the same proportion, SAS confirms similarity.
Worked example – using the SSS criterion
Consider triangle MNO with sides 5 cm, 7 cm, 9 cm and triangle RST with sides 10 cm, 14 cm, 18 cm. Each side of RST is exactly twice the matching side of MNO, so the ratio is 2:1 for all three pairs. By SSS, the triangles are similar.
Quick comparison of the three criteria
| Criterion | What you need | Why it works |
|---|---|---|
| AA | Two equal angles | Third angles must match automatically |
| SAS | One equal angle + proportion of the two sides forming that angle | Same shape forced by matching angle and side stretch |
| SSS | All three sides in the same proportion | Equal side ratios lock the shape, angles follow |
Step‑by‑step flow for proving similarity
Common pitfalls to avoid
- Mixing up corresponding sides with any random side. Always match the side opposite the equal angle.
- For SAS, the proportion must involve the sides that *sandwich* the equal angle, not any other side.
- Never assume similarity just because two triangles look alike on paper – you need a formal check.
📝 Likely Exam Questions
- Question: In triangle ABC, ∠A = 30°, ∠B = 60°, and AB = 5 cm. Triangle DEF has ∠D = 30°, ∠E = 60°, and DE = 10 cm. Prove the triangles are similar and find the length of DF if CF = 8 cm.
Answer: AA criterion gives similarity. Scale factor = 10/5 = 2. Hence DF = 2 × CF = 16 cm. - Question: Triangle PQR has sides 7 cm, 9 cm and angle Q = 45°. Triangle XYZ has sides 14 cm, 18 cm and angle Y = 45°. Show that the triangles are similar and calculate the length of XZ if PR = 12 cm.
Answer: Ratio of sides around the equal angle: 7/9 = 14/18 = 7/9. SAS criterion applies, so k = 14/7 = 2. Therefore XZ = 2 × PR = 24 cm. - Question: Two triangles have sides in the ratio 3:5:7 and 6:10:14 respectively. Are they similar? Justify.
Answer: All three side ratios are equal (6/3 = 10/5 = 14/7 = 2). By SSS criterion, the triangles are similar.