SSC CGL 2025 Tier-I · 19 September 2025, Shift 3 · Quantitative Aptitude · Question 20 of 100

We transcribed this text from the pictures on a candidate's response sheet for this shift; the answer is the key SSC colour-marked on that sheet. Solution last reviewed 11 September 2026.

Where this paper and its answers come from

SSC does not publish CGL question papers or a public answer key. From 2025 the candidate response sheet prints every question and option as a picture, so the text here is transcribed from those pictures for this shift and checked against them independently; where the layout carries meaning, the picture is shown as well. Each answer is the key SSC colour-marked on that sheet. SSC released a tentative key for challenge and later its final one, both only to candidates behind a login, and the sheet does not say which of the two it carries. The solutions are written by UnlockIAS.

In △ABC, an angle bisector from A meets BC at D. If AD bisects ∠BAC, and AB=AC, are △ABD and △ACD congruent? If so, by what rule?

(a)Yes, by SSS
(b)Yes, by SAS
(c)Yes, by ASA
(d)No, they are not congruent

Answer

Why

Correct — B. Match △ABD with △ACD using only what the stem gives.

Side: AB = AC (given)
Angle: ∠BAD = ∠CAD (AD bisects ∠BAC)
Side: AD = AD (common to both triangles)

The equal angle at A sits between the two pairs of equal sides, so it is the included angle.

Two sides and the included angle → SAS → option (b)

Why the others are wrong

(a)
Yes, by SSS — SSS needs BD = DC as a third pair of equal sides. The stem does not give it; it follows from the congruence, or from the angle-bisector theorem.
(c)
Yes, by ASA — ASA needs a second pair of equal angles. The stem gives one pair, at A. ∠B = ∠C comes from the isosceles-triangle theorem, which the stem does not state.
(d)
No, they are not congruent — Not congruent is ruled out: two pairs of sides and the angle between them match, and SAS guarantees congruence.

Concept

The triangle congruence rules are SSS, SAS, ASA, AAS and RHS. In SAS the equal angle must lie between the two equal sides; two sides and an angle that is not between them (SSA) do not guarantee congruence.

An angle bisector supplies one pair of equal angles, and a side shared by both triangles is always a free pair of equal sides.

Strictly, ASA and SSS can also be reached here, each with one extra theorem: equal base angles of an isosceles triangle give ∠B = ∠C, and the angle-bisector theorem gives BD ⁄ DC = AB ⁄ AC = 1.

The question asks which rule the given facts support directly, and that is SAS, as keyed.

Key facts

SAS: two sides and the angle between them are equal in both triangles.
SSA, two sides and a non-included angle, is not a congruence rule.
A side common to two triangles counts as one pair of equal sides.
In an isosceles triangle the bisector of the apex angle is also the median and is perpendicular to the base.

Study next

RHS congruence for right trianglesAngle-bisector theorem: BD ⁄ DC = AB ⁄ ACProperties of isosceles triangles: equal base angles, and the apex bisector as median and altitude

Common traps

Picking SSS by assuming BD = DC, which is a result of the congruence, not a given
Reaching for ASA through ∠B = ∠C, which is true here but is not among the facts the stem supplies

The same frame and the same four options appear at 19 Sep 2025, 09:00, Quant Q.20: AB = CD, ∠ABD = ∠CDB and the common diagonal BD, keyed (b) SAS. 12 Sep 2024, 09:00, Quant Q.17 asks which extra equality makes two triangles congruent by SAS.

Related PYQs

No directly related past PYQ was found.