SSC CGL 2024 Tier-I · 12 September 2024, Shift 3 · Quantitative Aptitude · Question 8 of 100

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 6 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.

In ΔABD and ΔFEC, ∠BAD = 60°, l(BD) = l(EC), ∠ABD = ∠FEC = 90°, and l(AB) = l(FE). Find the ratio of ∠BAD⁄∠FCE.

(a)2 : 3
(b)1 : 2
(c)2 : 5
(d)2 : 1

Answer

Why

Correct — D. The stem is an image. It reads: in △ABD and △FEC, ∠BAD = 60°, l(BD) = l(EC), ∠ABD = ∠FEC = 90° and l(AB) = l(FE); find ∠BAD ⁄ ∠FCE. The figure shows B, C, D, E on one horizontal line, A above B, F above E, and the dashed segments AD and FC crossing between them.

Rule: SAS congruence — legs AB = FE, legs BD = EC, and the equal included right angles at B and E.

So △ABD ≅ △FEC under A ↔ F, B ↔ E, D ↔ C.

In △ABD: ∠ADB = 180° − 90° − 60° = 30°.
The correspondence maps ∠ADB onto ∠FCE, so ∠FCE = 30°.

∠BAD ⁄ ∠FCE = 60 ⁄ 30 = 2 : 1 → option (d).

Why the others are wrong

(a)
2 : 3 — 2 : 3 would put ∠FCE at 90°, impossible in a triangle whose angle at E is already 90°. The third angle of △FEC is 30°.
(b)
1 : 2 — 1 : 2 is the ratio inverted. The question asks ∠BAD over ∠FCE, so 60° goes on top of 30°.
(c)
2 : 5 — 2 : 5 would need ∠FCE = 150°, more than the 90° left over once ∠FEC takes its right angle. Congruence pins ∠FCE at ∠ADB = 30°.

Concept

Two right triangles with one pair of equal legs, a second pair of equal legs and the equal right angle between them are congruent by SAS. Congruence then transfers every corresponding angle, which is all this question needs — no side length is given and none is required.

The correspondence is set by the given equalities, not by the letters' alphabetical order: AB = FE pairs A with F and B with E, and BD = EC pairs D with C.

Once the correspondence is written down, ∠FCE is simply ∠ADB, and in a right triangle with a 60° acute angle the other acute angle is 30°.

The figure draws the two triangles pointing in opposite directions with their bases on a shared line, which is what makes the correspondence easy to get wrong. Nothing in the figure is measured, so treat it as a labelling aid rather than a source of values.

Key facts

SAS congruence: two sides and the angle between them equal makes two triangles congruent.
In a right triangle with one acute angle of 60°, the remaining acute angle is 30°.
Corresponding angles of congruent triangles are equal, so ∠ADB = ∠FCE under A ↔ F, B ↔ E, D ↔ C.
Here ∠BAD = 60° and ∠FCE = 30°, so the ratio is 2 : 1.

Study next

SAS, ASA, SSS and RHS congruence and the correspondence each fixesCPCT and what it does and does not let you concludeAngles in a 30-60-90 right triangle and the side ratio that goes with them

Common traps

Pairing ∠FCE with ∠BAD because both triangles contain a 60° angle, which reads the correspondence backwards.
Looking for side lengths in the figure when the ratio needs only angles.
Answering 1 : 2 by writing the smaller angle first.

SSC hides a single congruence step behind a figure with two triangles facing opposite ways, so the mark goes to whoever writes the vertex correspondence down before answering. Circle geometry with a tangent and a secant is asked at Quant Q.1.

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