Triangles: The Magic of Similarity
Master similar triangles, the Basic Proportionality Theorem, and the criteria (AA, SSS, SAS) that power Class 10 geometry proofs and problems.
By the PadhoDost Team · 📖 7 min read · Updated 4 August 2026
Part of Class 10 (CBSE) prep🧠 The photocopy machine
Imagine you put a triangular sticker on a photocopier and press 'enlarge to 150%'. The copy is bigger, but it is the exact same shape - every angle is unchanged and every side has grown by the same factor. That is the whole idea of similar triangles: same shape, different size. Two triangles are similar when one is just a scaled-up or scaled-down photocopy of the other.
In this chapter we move from 'congruent' triangles (same shape AND same size) to 'similar' triangles (same shape, possibly different size). Similarity unlocks a huge range of real problems - finding the height of a tree from its shadow, the width of a river you cannot cross, or a distance on a map.
When are two triangles similar?
Two triangles are similar if their corresponding angles are equal AND their corresponding sides are in the same ratio (proportion). If triangle ABC is similar to triangle PQR, we write it as triangle ABC ~ triangle PQR, and the order of letters tells you which vertices match.
The Basic Proportionality Theorem (Thales' Theorem)
This is the star result of the chapter. If a line is drawn parallel to one side of a triangle and it cuts the other two sides, then it divides those two sides in the same ratio.
| Criterion | What you need to prove similarity |
|---|---|
| AA (or AAA) | Two angles of one triangle equal two angles of the other |
| SSS | All three pairs of corresponding sides in the same ratio |
| SAS | One pair of angles equal AND the two sides around that angle in the same ratio |
📝 Finding a length using BPT
In triangle ABC, DE is parallel to BC. D lies on AB with AD = 3 cm and DB = 5 cm. E lies on AC with AE = 4.5 cm. Find EC.
By the Basic Proportionality Theorem: AD/DB = AE/EC.
Substitute the values: 3/5 = 4.5/EC.
Cross-multiply: 3 x EC = 5 x 4.5 = 22.5.
So EC = 22.5 / 3 = 7.5 cm.
Quick revision
- ✓Similar = same shape, sides in equal ratio, angles equal. Congruent = similar AND same size.
- ✓Write similarity in matching vertex order, e.g. ABC ~ PQR.
- ✓Basic Proportionality Theorem: a line parallel to one side divides the other two sides proportionally.
- ✓Three similarity criteria: AA, SSS, SAS.
- ✓The ratio of corresponding sides of similar triangles is called the scale factor.
⚡ Quick check
In triangle ABC, DE is parallel to BC. If AD = 2 cm, DB = 3 cm and AE = 4 cm, what is EC?
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