MISSION · Geometry
When Two Triangles Must Be Identical
How little you need to know before the whole triangle is fixed
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The idea in one sentence
Three well-chosen measurements are enough to pin a triangle down completely, so two triangles sharing them must be the same triangle in a different position.
Congruent means identical
Two shapes are congruent if one can be picked up, turned, slid or flipped onto the other and land exactly. Every side matches a side, every angle matches an angle. Congruence is about equality; similarity is about shape — a scaled copy is similar but not congruent.
A triangle has six measurements: three sides and three angles. The surprise is that you never need all six. Three of the right kind do it.
The rules that work
SSS — three sides. Cut three sticks of fixed length and there is only one triangle they can make. Nothing can flex, which is why triangles are used for bracing and quadrilaterals aren’t.
SAS — two sides and the angle between them. Fix the angle, fix the two arms, and the third side has no choice about where to land.
ASA — two angles and the side between them. Draw the side, draw the two angles off its ends, and the arms meet at exactly one point.
AAS — two angles and a side not between them. This is ASA in disguise: the third angle is forced, because the angles sum to 180°, so you know the angle at the other end too.
RHS — for right triangles, the hypotenuse and one other side. The Pythagorean theorem fixes the remaining side, so it’s really SSS.
The two that fail
AAA fixes the shape, not the size. Two triangles with the same three angles are similar — one is a scaled photograph of the other. A 3-4-5 triangle and a 6-8-10 triangle have identical angles and are plainly not identical triangles.
SSA — two sides and an angle that isn’t between them — usually fails, and the way it fails is worth seeing. Fix the angle and one arm, then swing the second side down. If it’s short enough, it can reach the base line at two different points, giving two genuinely different triangles from the same three measurements. This is the ambiguous case, and it comes back later as the reason the sine rule can produce two answers.
Using it
The point of the rules is proof. To show two lengths are equal, find triangles containing them, show the triangles are congruent by one of the rules, and conclude that the matching parts are equal. That last step gets its own name — corresponding parts of congruent triangles are equal — and it’s where the actual result comes from.
Shared sides and vertical angles are the usual free gifts: a side both triangles contain is equal to itself, and crossing lines hand you a pair of equal angles.
Common mix-ups
- Using SSA and calling it SAS. The angle must be between the two sides.
- Claiming congruence from AAA. Same shape, any size.
- Mismatching the order of the letters when writing △ABC ≅ △DEF, which then claims the wrong parts are equal.
- Forgetting the shared side in a proof, then being one fact short.
Check yourself
Two sides and the angle between them are equal — which rule? And does having all three angles equal make two triangles congruent?
Two sides with the included angle is SAS. Three equal angles give similarity only: the triangles have the same shape, but either could be any size.
LEVEL 3 · REFLECT
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