Ever noticed how a tiny map can show the whole city? That's the magic of similar triangles – they keep the same shape even when they’re bigger or smaller.
💡 In Simple Words: Two triangles are similar when their angles match exactly and their sides grow or shrink by the same factor. Think of a photo that you zoom in on – the picture stays the same, just bigger.
What is Similarity of Triangles?
In geometry, similarity means two shapes have the same shape but not necessarily the same size. For triangles, this means all three angles are equal, and the corresponding sides are in proportion – like the ratio stays constant.
Key Theorems for Similar Triangles
ICSE expects you to know three main ways to prove two triangles are similar:
- AA (Angle‑Angle) Theorem
- SSS (Side‑Side‑Side) Theorem
- SAS (Side‑Angle‑Side) Theorem
Let’s break each one down.
AA (Angle‑Angle) Theorem
The AA theorem says: If two angles of one triangle are equal to two angles of another triangle, the triangles are similar. You don’t even need to check the sides – the angles do all the work.
Why does it work? A triangle’s three angles always add up to 180°. If you know two angles match, the third one must match automatically. It’s like knowing two pieces of a puzzle; the last piece is forced to fit.
SSS (Side‑Side‑Side) Theorem
The SSS theorem states: If the three sides of one triangle are in the same ratio as the three sides of another triangle, the triangles are similar. Here, you compare side lengths, not angles.
Imagine you have two ladders that are exactly the same shape but one is twice as tall. Every rung (side) is twice as long, so the ladders are similar.
SAS (Side‑Angle‑Side) Theorem
The SAS theorem says: If two sides of one triangle are in proportion to two sides of another triangle and the included angle (the angle between those sides) is equal, the triangles are similar.
Think of two slices of pizza that have the same crust‑to‑tip ratio and the same tip angle – they will look alike no matter how big each slice is.
Worked Example
Let’s solve a typical ICSE problem.
Problem: In triangle ABC, AB = 6 cm, AC = 9 cm, and ∠A = 60°. Triangle DEF has DE = 4 cm, DF = 6 cm, and ∠D = 60°. Prove that the two triangles are similar and find the ratio of their areas.
Solution:
- Notice that ∠A = ∠D = 60° – the included angles are equal.
- Check the sides around those angles: AB/DE = 6/4 = 3/2 and AC/DF = 9/6 = 3/2. The two ratios are the same.
- Since two sides are in the same proportion and the included angle is equal, by the SAS theorem, ΔABC ∼ ΔDEF.
- The similarity ratio (scale factor) is 3/2 (big triangle : small triangle). Area scales by the square of the ratio, so area ratio = (3/2)² = 9/4.
So the larger triangle’s area is 9/4 times the smaller one.
Quick Summary Table
| Theorem | What You Need | Result |
|---|---|---|
| AA | Two equal angles | Triangles are similar |
| SSS | All three side ratios equal | Triangles are similar |
| SAS | Two side ratios equal + included angle equal | Triangles are similar |
📝 Likely Exam Questions
- Question: In ΔPQR, PQ = 8 cm, PR = 12 cm, and ∠P = 45°. In ΔXYZ, XY = 4 cm, XZ = 6 cm, and ∠X = 45°. Prove similarity and find the ratio of perimeters.
Answer: By SAS (ratio 8/4 = 2, 12/6 = 2, and ∠P = ∠X), ΔPQR ∼ ΔXYZ with scale factor 2. Perimeter ratio = 2:1. - Question: Two triangles have angles 30°, 60°, 90° and 30°, 60°, 90°. Are they similar? Explain.
Answer: Yes, by AA – two angles match, the third must match, so the triangles are similar. - Question: If the sides of ΔMNO are in the ratio 5:7:9 and ΔRST has sides 10 cm, 14 cm, 18 cm, are the triangles similar?
Answer: All three side ratios are 2 (10/5, 14/7, 18/9). By SSS, ΔMNO ∼ ΔRST. - Question: In ΔABC, AB = 5 cm, BC = 7 cm, AC = 9 cm. In ΔDEF, DE = 10 cm, EF = 14 cm, DF = 18 cm. Find the ratio of their areas.
Answer: All side ratios are 2, so by SSS the triangles are similar. Area ratio = 2² = 4:1.