SSC CGL 2024 Tier-I · 10 September 2024, Shift 1 · Quantitative Aptitude · Question 3 of 98

Text as printed on a candidate's response sheet for this shift; answer per SSC's tentative key as marked on that sheet. Solution last reviewed 10 September 2026.

Where this paper and its answers come from

SSC does not publish CGL question papers or a public answer key. The questions and options here are reproduced as printed on a candidate's own response sheet for this shift — the Commission's text, unchanged. Each answer is SSC's tentative key as marked on that sheet; SSC allowed challenges to it and never publishes a final key. The solutions are written by UnlockIAS.

By applying which of the following criteria can two triangles NOT be proved congruent?

(a)side-side-side
(b)angle-angle-angle
(c)side-angle-side
(d)angle-side-angle

Answer

Why

Correct — B. Angle-angle-angle fixes shape, not size.

Two triangles with the same three angles are similar: their sides are in a fixed ratio, but that ratio can be anything at all. A 3-4-5 triangle and a 6-8-10 triangle have identical angles and are plainly not congruent.

Congruence needs at least one side, because a side is the only element that carries a length and so sets the scale. The standard criteria — SSS, SAS, ASA, AAS and RHS — each contain one. AAA does not, so option (b) is the criterion that cannot prove congruence.

Why the others are wrong

(a)
side-side-side — side-side-side is a valid criterion. Three side lengths determine a triangle completely, so two triangles with all sides equal must be congruent.
(c)
side-angle-side — side-angle-side is valid. Two sides and the angle between them fix the third side, so nothing about the triangle is left free.
(d)
angle-side-angle — angle-side-angle is valid. Two angles force the third, and the one known side then fixes the scale that AAA leaves open.

Concept

Congruent triangles agree in shape and size. Similar triangles agree only in shape.

Every congruence criterion contains at least one side: SSS, SAS, ASA, AAS and RHS all do. AAA contains none, so it can only establish similarity — the two triangles may differ by any scale factor.

The point is worth holding in the other direction as well. AAA similarity plus a single pair of equal corresponding sides pins the scale factor at 1 and upgrades the pair to congruent.

The stem asks which criterion can NOT prove congruence, so the odd one out is the answer. Read the negative wording before scanning the options, because three of the four are perfectly valid.

Key facts

The triangle congruence criteria are SSS, SAS, ASA, AAS and RHS.
AAA establishes similarity only, never congruence.
Similar triangles have equal corresponding angles and proportional sides.
RHS applies to right triangles, matching the hypotenuse and one other side.

Study next

AAA and SSS similarity and the ratio of areas of similar trianglesWhy SSA fails as a congruence criterionUsing CPCT to extract equal parts once congruence is proved

Common traps

Missing the word NOT and choosing a criterion that does work.
Confusing AAA with AAS, which is valid precisely because it carries a side.

Congruence is set in several framings. At 09 Sep 2024, 09:00, Quant Q.5 the question is which statement is sufficient for congruence, and at 12 Sep 2024, 09:00, Quant Q.17 you are given AB = FD and ∠A = ∠D and asked which further equality completes SAS.

Related PYQs

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