Triangle Congruence Worksheet
Practise triangle congruence on one free printable worksheet. It covers naming the right postulate from the given parts, working out what further information would complete a proof, deciding whether two triangles must be congruent, and using CPCTC correctly.
Download the worksheet
Four pages of practice and a three-page worked answer key, with the reasoning spelled out for each.
What is on the worksheet
- Section A — name the postulate (8 questions). From the given congruent parts, including one that is not enough and one HL case.
- Section B — what else do you need? (6 questions). Each pair is one fact short of a proof.
- Section C — must they be congruent? (6 questions). Yes or no, with a postulate or a counter-example.
- Section D — explain your reasoning (2 questions). Why HL works when SSA does not, and why CPCTC never proves congruence.
- Answer key (3 pages). Each postulate named with the reason it applies.
The five that work, and the two that do not
Five combinations of parts are enough to prove two triangles congruent. Two that look similar are not.
- SSS — all three pairs of sides.
- SAS — two sides and the angle between them.
- ASA — two angles and the side between them.
- AAS — two angles and a side that is not between them.
- HL — right triangles only: the hypotenuse and one leg.
- Not postulates: SSA can describe two different triangles, and AAA fixes the shape but not the size.
Worked example: Given two sides and an angle, the position of the angle decides everything. Between the two sides it is SAS and the triangles are congruent. Opposite one of them it is SSA, and two different triangles can fit the same three facts.
Why HL works when SSA does not
HL is a right triangle with two sides and a non-included angle — which looks exactly like the SSA case that fails in general. The right angle is what rescues it.
In a right triangle the right angle is the largest, so every other angle must be acute, and the ambiguity disappears. With the hypotenuse and one leg known, Pythagoras forces the third side — which makes HL really SSS in disguise.
Using CPCTC the right way round
CPCTC stands for “corresponding parts of congruent triangles are congruent”. It says that once two triangles are known to be congruent, every pair of corresponding parts is congruent too.
It is a consequence of congruence, never evidence for it. Prove congruence first with SSS, SAS, ASA, AAS or HL, and only then use CPCTC for the remaining parts. Using it inside a proof assumes what you are trying to show.
The mistake to watch for
Writing SAS when the angle is not between the two sides. The order of the letters is the whole point: S-A-S means side, included angle, side. If the angle sits outside the pair, you have SSA and no proof.
Frequently asked questions
What are the five triangle congruence postulates?
SSS, SAS, ASA, AAS and HL. HL applies only to right triangles.
Why is SSA not a congruence postulate?
Because two sides and a non-included angle can describe two different triangles. The angle opposite a known side could be acute or obtuse, and both may give a valid triangle — the same ambiguity that appears in the law of sines.
Is AAA a congruence postulate?
No. Three equal angles make triangles similar, not congruent. They have identical shapes but can be any size. One pair of corresponding sides is needed to fix the scale.
What does CPCTC mean?
Corresponding parts of congruent triangles are congruent. It is used after a congruence proof to justify claims about the remaining sides and angles, never as a reason within the proof itself.