Similarity criteria — AA, SAS, SSS for triangles with proofs

Question

What are the three criteria for similarity of triangles — AA, SAS, and SSS? How is similarity different from congruence? State and apply the Basic Proportionality Theorem (Thales’ theorem).

(CBSE Class 10 — Triangles chapter carries 10-15 marks in boards; includes proofs and applications)

Solution — Step by Step

If two angles of one triangle are equal to two angles of another triangle, the triangles are similar. (The third angle is automatically equal since the angle sum is 180°.)

This is the most commonly used criterion. We write: $\triangle ABC \sim \triangle DEF$ .

Similar means: same shape, possibly different size. Corresponding sides are proportional.

If one angle of a triangle is equal to one angle of another, and the sides including these angles are proportional, then the triangles are similar.

\frac{AB}{DE} = \frac{AC}{DF} \quad \text{and} \quad \angle A = \angle D \implies \triangle ABC \sim \triangle DEF

Note: for similarity, we need the ratio of sides to be equal, not the sides themselves (that would be congruence).

If all three pairs of corresponding sides are proportional, the triangles are similar.

\frac{AB}{DE} = \frac{BC}{EF} = \frac{AC}{DF} \implies \triangle ABC \sim \triangle DEF

This ratio is the scale factor of the similarity.

If a line is drawn parallel to one side of a triangle, it divides the other two sides proportionally.

If $DE \parallel BC$ in $\triangle ABC$ :

\frac{AD}{DB} = \frac{AE}{EC}

This theorem is the foundation for most similarity proofs in CBSE Class 10.

flowchart TD
    A["Are two triangles similar?"] --> B{What is given?}
    B -->|Two angles equal| C["AA Similarity"]
    B -->|One angle equal + including sides proportional| D["SAS Similarity"]
    B -->|All three sides proportional| E["SSS Similarity"]
    A --> F["If similar:<br/>Corresponding sides proportional<br/>Corresponding angles equal"]
    F --> G["Area ratio = (side ratio)²"]

Why This Works

Similarity is a weaker condition than congruence. Congruent triangles are identical in shape AND size. Similar triangles have the same shape but can be different sizes — one is a scaled version of the other.

The reason only two angles are needed (AA) is that the third angle is fixed by the angle sum property. And once the shape is fixed (all angles determined), the sides must be in a fixed ratio — there is no freedom left.

The area ratio result is particularly useful: if two similar triangles have sides in the ratio $k$ , their areas are in the ratio $k^2$ . This appears frequently in CBSE board questions.

Alternative Method

For board exams, the most important application of BPT is the midpoint theorem: a line joining the midpoints of two sides of a triangle is parallel to the third side and half its length. This is BPT in reverse — if the sides are divided equally ( $AD/DB = AE/EC = 1$ ), then $DE \parallel BC$ and $DE = BC/2$ .

Common Mistake

Students confuse the order of vertices in similarity notation. When we write $\triangle ABC \sim \triangle DEF$ , it means $A$ corresponds to $D$ , $B$ to $E$ , $C$ to $F$ . So $AB/DE = BC/EF = AC/DF$ . Writing the triangles in the wrong order gives wrong proportions. Always match the vertices carefully: the vertex opposite the longest side in one triangle must correspond to the vertex opposite the longest side in the other.