Similar triangles — AA, SAS, SSS criteria with proof and problems

Question

In $\triangle ABC$ , $DE \parallel BC$ where $D$ is on $AB$ and $E$ is on $AC$ . If $AD = 4$ cm, $DB = 6$ cm, and $AE = 3$ cm, find $EC$ . Also, state and prove the AA criterion for similarity.

(NCERT Class 10 — Triangles, CBSE Board favourite)

Solution — Step by Step

Since $DE \parallel BC$ , by the Basic Proportionality Theorem (BPT / Thales’ theorem), $\triangle ADE \sim \triangle ABC$ . When a line is drawn parallel to one side of a triangle, it creates a smaller triangle similar to the original.

By BPT: $\frac{AD}{DB} = \frac{AE}{EC}$

\frac{4}{6} = \frac{3}{EC}

EC = \frac{3 \times 6}{4} = \frac{18}{4} = \mathbf{4.5 \text{ cm}}

Proof of AA Criterion

AA Criterion: If two angles of one triangle are equal to two angles of another triangle, the triangles are similar.

Given: In $\triangle ABC$ and $\triangle DEF$ , $\angle A = \angle D$ and $\angle B = \angle E$ .

To prove: $\triangle ABC \sim \triangle DEF$ .

Proof: Since $\angle A = \angle D$ and $\angle B = \angle E$ , we get $\angle C = 180° - \angle A - \angle B = 180° - \angle D - \angle E = \angle F$ .

All three angles are equal. By the definition of similar triangles (same shape, proportional sides), $\triangle ABC \sim \triangle DEF$ and therefore:

\frac{AB}{DE} = \frac{BC}{EF} = \frac{AC}{DF}

Why This Works

Similarity means same shape but possibly different size. Two triangles are similar when their corresponding angles are equal AND corresponding sides are proportional. The three criteria provide shortcuts:

AA: Two angles matching forces the third to match (angle sum = 180°), which guarantees proportional sides.
SSS Similarity: If $\frac{AB}{DE} = \frac{BC}{EF} = \frac{AC}{DF}$ , then corresponding angles must be equal.
SAS Similarity: If $\frac{AB}{DE} = \frac{AC}{DF}$ and $\angle A = \angle D$ , then the triangles are similar.

The BPT result used in the numerical is itself a consequence of AA similarity — the parallel line creates equal corresponding angles.

Alternative Method

You could also use the ratio $\frac{AD}{AB} = \frac{AE}{AC}$ instead of $\frac{AD}{DB} = \frac{AE}{EC}$ . Here, $AB = AD + DB = 10$ , so $\frac{4}{10} = \frac{3}{AC}$ , giving $AC = 7.5$ cm and $EC = AC - AE = 7.5 - 3 = 4.5$ cm. Same answer, slightly different route.

In CBSE 10th board, similarity questions carry 3-5 marks. The proof of BPT or AA criterion is a guaranteed question almost every year. Practise writing the proof in under 5 minutes — it saves time for later questions.

Common Mistake

Students often confuse congruence criteria (SSS, SAS, ASA) with similarity criteria (AA, SSS ratio, SAS ratio). For congruence, sides must be EQUAL. For similarity, sides must be PROPORTIONAL. Writing “SSS congruence” when the question asks about similarity will cost you marks.