Criteria for similarity (AAA SSS SAS)

Checking every angle and every side ratio to prove two triangles are similar is more work than needed. Just as the Class 9 congruence criteria let you certify "identical" triangles from partial data, three similarity criteria — AAA (or AA), SSS, and SAS — let you certify "same shape" triangles from partial data too. This is the central toolkit of the CBSE Class 10 chapter Triangles, built directly on the ideas of similar figures and similarity of triangles. A student is expected to state each criterion, recognise which one applies from the data given in a figure, and use it to conclude similarity and extract unknown side lengths or angles. The current CBSE syllabus treats these as statement and use — formal proofs are de-emphasised, so exam focus falls on application: spotting the right criterion fast and reading off the correct vertex correspondence.

The core insight

Two triangles are similar — same shape, angles equal, sides proportional — the moment any one of AA (two equal corresponding angles), SSS (all three sides proportional), or SAS (one equal included angle plus its two flanking sides proportional) is verified from the given data.

The core idea

Three shortcuts, each sufficient on its own to prove two triangles similar without checking every angle and every side.

AAA (or AA) similarity criterion. If in two triangles corresponding angles are equal, the triangles are similar. Since the three angles of any triangle sum to $180^\circ$ , matching just two pairs of corresponding angles forces the third pair to match automatically — so in practice this criterion is applied as AA:

\angle A=\angle D,\ \angle B=\angle E \implies \triangle ABC \sim \triangle DEF

SSS similarity criterion. If in two triangles corresponding sides are all in the same ratio, the triangles are similar. This differs from SSS congruence, which needs the sides equal, not merely proportional:

\frac{AB}{DE}=\frac{BC}{EF}=\frac{CA}{FD} \implies \triangle ABC \sim \triangle DEF

SAS similarity criterion. If one pair of corresponding angles is equal and the two sides including that angle are in the same ratio, the triangles are similar:

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

where: $\angle A, \angle B, \angle D, \angle E$ are corresponding angles of the two triangles, and $AB, BC, CA, DE, EF, FD$ are the corresponding sides being compared. A quick way to remember the difference from Class 9 congruence: congruence criteria (SSS, SAS, ASA, RHS) require sides to be equal; similarity criteria require sides to be in the same ratio — congruence is just the special case $k=1$ of similarity, where the scale factor is exactly one. Once any single criterion establishes $\triangle ABC \sim \triangle DEF$ , all the similarity facts follow at once — every corresponding angle equal, every corresponding side proportional — even though only a subset of the data was checked to get there.

Figure

Three pairs of triangles side by side illustrating the AA SSS and SAS similarity criteria with matching angle marks and side ratio labels

Derivation AWhy two equal angles are always enough to prove AAA

The goal is to show, using only the angle-sum property of a triangle that every CBSE Class 9 student already owns, that checking just two pairs of corresponding angles in two triangles automatically forces the third pair to be equal too — which is exactly why the AAA criterion is applied in practice as the lighter AA test, without ever measuring a third angle.

Step 1. Write the angle-sum property for both triangles. In $\triangle ABC$ , $\angle A + \angle B + \angle C = 180^\circ$ . In $\triangle DEF$ , $\angle D + \angle E + \angle F = 180^\circ$ . Both totals equal the same constant, $180^\circ$ , because this holds for every triangle.

Step 2. Substitute the two given equalities. Suppose the data gives $\angle A = \angle D$ and $\angle B = \angle E$ . Replace $\angle D$ and $\angle E$ in the second equation with $\angle A$ and $\angle B$ : $\angle A + \angle B + \angle F = 180^\circ$ .

Step 3. Compare the two rewritten sums. From Step 1, $\angle A + \angle B + \angle C = 180^\circ$ ; from Step 2, $\angle A + \angle B + \angle F = 180^\circ$ . Both left-hand sides share the same $\angle A + \angle B$ and equal the same total, so subtracting gives $\angle C = \angle F$ .

Step 4. Read the result. The third pair of angles is equal without any separate measurement — it is forced by algebra, not a new geometric fact. This is exactly why the similarity criterion is stated as AAA but tested as AA: verifying two angle pairs already certifies all three, and once all three angles match, the triangles are similar by definition of the AAA criterion.

Worked example

In $\triangle PQR$ , $PQ = 4\text{ cm}$ , $QR = 6\text{ cm}$ , $\angle Q = 80^\circ$ . In $\triangle LMN$ , $LM = 6\text{ cm}$ , $MN = 9\text{ cm}$ , $\angle M = 80^\circ$ . Determine whether $\triangle PQR \sim \triangle LMN$ , find the correct correspondence, and hence find $PR:LN$ .

Step 1. Identify which criterion the data fits. Two sides and the angle between them are given in each triangle, so check the SAS similarity criterion: the included angle of one triangle must equal the included angle of the other, and the two pairs of sides forming that angle must be in the same ratio.

Step 2. Check the angle. $\angle Q$ is included between sides $QP$ and $QR$ ; $\angle M$ is included between sides $ML$ and $MN$ . Both are given as $80^\circ$ , so $\angle Q = \angle M$ .

Step 3. Check the side ratios, matching $Q \leftrightarrow M$ .

\frac{QP}{ML}=\frac{4}{6}=\frac{2}{3}, \qquad \frac{QR}{MN}=\frac{6}{9}=\frac{2}{3}

Both ratios equal $\dfrac{2}{3}$ , and the equal angle sits correctly between the two compared sides in both triangles, so the SAS test is satisfied.

Step 4. State the conclusion and correspondence. By the SAS similarity criterion, $\triangle PQR \sim \triangle MLN$ , with correspondence $P \leftrightarrow L$ , $Q \leftrightarrow M$ , $R \leftrightarrow N$ . Since all corresponding sides of similar triangles carry the same ratio as the pair already verified, $\dfrac{PR}{LN} = \dfrac{2}{3}$ , so $PR : LN = 2 : 3$ .

Competitive stretch

If it is further given that the perimeter of $\triangle LMN$ is $18\text{ cm}$ , the same scale factor $\dfrac{2}{3}$ (this time read as $PQR:LMN$ ) gives the perimeter of $\triangle PQR$ directly: perimeter of $\triangle PQR = \dfrac{2}{3} \times 18 = 12\text{ cm}$ — no need to find every individual side first, since the ratio of perimeters of similar triangles always equals the ratio of any pair of corresponding sides.

Common traps

Applying SAS with the equal angle not included between the two proportional sides — the angle must sit between the two sides being compared, otherwise the criterion does not apply.
Confusing similarity's SSS/SAS (sides proportional) with congruence's SSS/SAS (sides equal) — writing "SSS implies congruent" when only ratios, not equal lengths, were verified.
Assuming AA needs all three angles checked — only two pairs of equal corresponding angles are needed; the third follows automatically from the angle-sum property.
Getting the correspondence wrong after proving similarity, then reading off an incorrect side ratio for an unknown length, as in the worked example above where $M$ , not $L$ , corresponds to $Q$ .

Ask the tutor: Given a figure with only some angles or sides marked, ask the AI tutor to help you decide which similarity criterion — AA, SSS, or SAS — actually applies to your specific data before committing to an answer.

Why is AAA usually written and used as just AA

Because the third angle is not independent information. Every triangle's angles sum to $180^\circ$ , so once two pairs of corresponding angles are shown equal, subtracting from $180^\circ$ on both sides forces the third pair to be equal too. Checking a third angle only confirms what algebra already guaranteed, so exam answers correctly state "AA similarity criterion" or "AAA similarity criterion" interchangeably — both mean the same test.

How is SSS similarity different from SSS congruence if both use all three sides

The difference is equal versus proportional. SSS congruence needs the three sides of one triangle to be exactly equal in length to the three sides of the other. SSS similarity only needs the three ratios of corresponding sides to be equal to each other, for example all three ratios equal to $\dfrac{2}{3}$ — the triangles can be different sizes and still pass. Congruence is the particular case where that common ratio happens to be $1$ .

How do I decide fast which criterion to use in an exam question

Look at what the figure or statement actually gives you. Three angles or two angles given means try AA. Three side lengths given, with none of the equal angles, means try SSS by computing all three ratios. Two sides plus the angle between exactly those two sides means try SAS. If the given angle is not sandwiched between the two given sides, none of these criteria directly applies from that data alone, so check the figure again before assuming similarity.

Criteria for similarity (AAA SSS SAS)

The core idea

Worked example

Competitive stretch

Common traps

Before this

Where this leads