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SSC CGL Maths · Triangles

Congruent Triangles

Congruent triangles are exact copies: the order of letters in the statement pairs their corners, and SSS, SAS, ASA or RHS is enough to prove it.

All Triangles notes

Why this matters

Thirty-two questions in the bank, 31 EASY and 1 MODERATE, the largest group in the chapter. Twelve hand you a statement like △ABC ≅ △RPQ and ask which side or angle matches; fifteen ask which rule applies, or whether one applies at all. Nearly every wrong option is a part matched by its position in the figure instead of by the order of the letters.

Concept 1 of 3: What congruent triangles have in common

Two congruent triangles are the same triangle placed in two spots: cut one out and it fits exactly on the other, perhaps after a turn or a flip. Anything you can measure on one, you get on the other.

Definition

  • Congruent (written ≅) means the same shape and the same size.
  • So all three pairs of matching sides are equal and all three pairs of matching angles are equal.
  • It follows that the perimeters are equal, the areas are equal, and the ratio of any side to its matching side is 1 : 1.
  • Two triangles with the same three side lengths are congruent, so they cannot have different areas or different angles.
  • Congruent triangles are always similar. Triangles with only the same angles have the same shape, but may differ in size.

Worked example

Triangle ABC has sides 9 cm, 12 cm and 15 cm, and triangle DEF is congruent to it. Find the perimeter and the area of triangle DEF.
Practice this conceptself-check · 4 quick reps

The same idea in a real exam question:

SSC CGL · 2025 · Tier I, 18 Sep 2025, Shift 2 · Q70Easy

Example 1 · Triangles · Congruent Triangles

△ABC\triangle ABC has sides AB = 5 cm, BC = 6 cm, AC = 7 cm. △XYZ\triangle XYZ is congruent to △ABC\triangle ABC. What is the perimeter of △XYZ\triangle XYZ?

Taking equal angles as congruence

Three equal angles fix only the shape of a triangle. A small and a large equilateral triangle have the same angles and are not congruent.

Believing congruent triangles can differ in area

Congruent triangles are exact copies, so any option saying their areas or perimeters differ is false.
Drill 4 more on what congruent triangles have in common

Concept 2 of 3: Reading the matching corners from a congruence statement

A congruence statement is written so that its letters line up: the first corner of one triangle goes with the first corner of the other, and so on. The statement, not the drawing, tells you which parts are equal.

Definition

  • In △ABC≅△PQR\triangle ABC \cong \triangle PQR the corners pair in the order written: A with P, B with Q, C with R.
  • Sides pair the same way: AB = PQ, BC = QR, CA = RP. Angles too: ∠A=∠P\angle A = \angle P, ∠B=∠Q\angle B = \angle Q, ∠C=∠R\angle C = \angle R.
  • Method: write the two names one under the other and read down the columns. To find the side matching PQ in △ABC≅△RPQ\triangle ABC \cong \triangle RPQ, see that P and Q are the 2nd and 3rd letters, so the answer is the 2nd and 3rd letters of ABC: BC.
  • When the parts are given as expressions, set the matching parts equal and solve.
  • When two angles of one triangle are known, the third comes from the angle sum, and it equals its partner in the other triangle.

Matching parts

△ABC≅△PQR  ⇒  AB=PQ,  BC=QR,  CA=RP,  ∠A=∠P,  ∠B=∠Q,  ∠C=∠R\triangle ABC \cong \triangle PQR \;\Rightarrow\; AB = PQ,\; BC = QR,\; CA = RP,\; \angle A = \angle P,\; \angle B = \angle Q,\; \angle C = \angle R

Worked example

△KLM≅△TRS\triangle KLM \cong \triangle TRS, with KL = 9 cm, LM = 12 cm, ∠K=58∘\angle K = 58^\circ and ∠L=71∘\angle L = 71^\circ. Find RS and ∠S\angle S.
Practice this conceptself-check · 4 quick reps

The same idea in a real exam question:

SSC CGL · 2024 · Tier I, 18 Sep 2024, Shift 1 · Q61Moderate

Example 2 · Triangles · Congruent Triangles

If △ABC\triangle ABC and △QPR\triangle QPR are congruent triangles such that AB = 7 cm, AC = 8 cm, ∠B=65∘\angle B = 65^\circ and ∠C=75∘\angle C = 75^\circ, then which of the following is true?

Matching corners by alphabet or by position

In △ABC ≅ △FDE, A matches F, B matches D and C matches E. The pairing comes only from the order the letters are written in.

Trusting how the figure is drawn

Two congruent triangles may be drawn turned or flipped. Read the matching sides from the statement, then check the figure against it.
Drill 11 more on reading the matching corners from a congruence statement

Concept 3 of 3: The congruence rules: SSS, SAS, ASA and RHS

You do not need all six parts to know two triangles are copies. Three well-chosen parts already fix a triangle completely, so if those three match, everything matches.

Definition

  • SSS: all three pairs of sides equal.
  • SAS: two sides and the angle between them (the included angle).
  • ASA: two angles and the side between them. AAS (two angles and any matching side) also works, because the third angle follows from the angle sum.
  • RHS: a right angle, the hypotenuse and one other side.
  • Not rules: AAA fixes only the shape, and SSA (an angle that is not between the two sides) does not fix the triangle, except when that angle is 90°, which is RHS.
  • Free parts: a side shared by both triangles, and vertically opposite angles where two lines cross, count as equal.
  • Two equal pairs of sides plus equal perimeters give the third pair equal, so SSS. The same base with the same area or height is not enough: the top corner can slide along a line.
  • Once two triangles are congruent, every matching part is equal. This is how a congruence proves two angles or sides equal.

Worked example

In quadrilateral PQRS the diagonal PR is drawn, PQ = RS and QR = SP. Are triangles PQR and RSP congruent? If so, by which rule, and which angle equals ∠PQR\angle PQR?
Practice this conceptself-check · 5 quick reps

The same idea in a real exam question:

SSC CGL · 2025 · Tier I, 19 Sep 2025, Shift 1 · Q70Easy

Example 3 · Triangles · Congruent Triangles

In a quadrilateral ABCD, a line segment BD is a diagonal such that AB=CD and ∠ABD=∠CDB\angle ABD = \angle CDB. Are the triangles ABD and CDB congruent? If so, by what rule?

Accepting two sides and a non-included angle

Two sides with an angle that is not between them (SSA) do not fix a triangle. The only exception is a right angle, which is the RHS rule.

Calling a right-angle case SAS

With a right angle, the hypotenuse and a leg equal, the right angle is not between the two given sides. The rule is RHS, not SAS.

Missing the shared side or the crossing angles

A side common to both triangles, or two vertically opposite angles, are equal parts that the question does not state. Count them.
Drill 14 more on the congruence rules: sss, sas, asa and rhs

Summary — formulas & gotchas at a glance

A revision cheat-sheet for the formulas and gotchas above. Click any concept name to jump back to its full explanation.

Formulas (1)

  • Reading the matching corners from a congruence statement

    Matching parts

    △ABC≅△PQR  ⇒  AB=PQ,  BC=QR,  CA=RP,  ∠A=∠P,  ∠B=∠Q,  ∠C=∠R\triangle ABC \cong \triangle PQR \;\Rightarrow\; AB = PQ,\; BC = QR,\; CA = RP,\; \angle A = \angle P,\; \angle B = \angle Q,\; \angle C = \angle R

Watch out for (7)

Test yourself on Triangles

20 past SSC CGL questions from this chapter, timed at 12 minutes and marked the way the exam marks it. You see your score and every answer the moment you finish. Free to start.

Take the Triangles chapter testDrill all 32 questions