Triangles • Topic 1 of 4

Congruence

Two triangles are congruent when they are identical in shape AND size — every pair of corresponding sides and angles matches. In CAT you almost never "prove" congruence formally; instead you use it as a tool to transfer a known length or angle from one triangle to an unknown one in the same figure. There are exactly four valid tests: SSS (all three sides equal), SAS (two sides and the INCLUDED angle), ASA (two angles and the included side) — and its cousin AAS — and RHS (right angle, hypotenuse and one side) for right triangles. The single most common trap is SSA, sometimes called the "ambiguous case": two sides and a non-included angle do NOT guarantee congruence. Note also that AAA only proves similarity, never congruence, because the triangles can scale. Once congruence is established, corresponding parts (CPCT) are equal, which is usually the fact the question secretly needs.

✅ Solved examples

1. In triangle ABC, AB = AC and AD is the median to BC. State the test that proves triangle ABD ≅ triangle ACD and the consequence.
AB = AC (given), BD = DC (median), AD = AD (common) ⇒ SSS. Hence by CPCT angle ADB = angle ADC, and as they are a linear pair each is 90°, so the median to the base of an isosceles triangle is also the altitude.
2. Two triangles have sides 6, 8 and included angle 50°, and 6, 8 with included angle 50°. Are they congruent? By which test?
Two sides and the angle BETWEEN them are equal, so SAS applies ⇒ the triangles are congruent.
3. A triangle has sides 7 and 9 with a 40° angle opposite the side of length 7. Is a triangle with the same data necessarily congruent to it?
The 40° is NOT between the two sides — this is the SSA / ambiguous case. Two non-congruent triangles can fit the data, so congruence is NOT guaranteed.
4. Right triangles PQR and XYZ have right angles at Q and Y, hypotenuses PR = XZ = 13 and legs PQ = XY = 5. Are they congruent, and what is the third side?
Right angle + equal hypotenuse + equal one side ⇒ RHS, so the triangles are congruent. The remaining leg = √(13² − 5²) = √(169 − 25) = √144 = 12.

✏️ Practice — try these, take hints as needed

1. Which single test proves congruence of two triangles given all three pairs of sides equal?
No angles are mentioned.
Side-Side-Side.
Abbreviate it.
SSS
2. Triangles with (5, 7, 60° included) and (5, 7, 60° included): congruent? Name the test.
The angle is between the two sides.
That is the included angle.
Side-Angle-Side.
Yes, by SAS
3. Does AAA (all three angles equal) prove two triangles congruent?
Equal angles fix shape only.
A small and a large copy share angles.
It proves similarity, not size.
No — AAA gives similarity only
4. In a right triangle, hypotenuse 25 and one leg 7. The other leg?
Use Pythagoras.
25² − 7² = 625 − 49.
√576.
24
5. Two triangles share two equal sides and a non-included equal angle. Is congruence guaranteed?
Identify the configuration.
This is SSA.
It is the ambiguous case.
No — SSA does not guarantee congruence

📝 Topic test — 8 questions

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