Congruence criteria for triangles

Builds on Congruent triangles Triangle Angle at a point Euclidean distance

Statement

Let $ABC$ and $A'B'C'$ be triangles in the Euclidean plane. Write $XY$ for the distance between points $X$ and $Y$ , and $\angle A$ for the angle $\angle BAC$ , and so on. The triangles are congruent under $A \mapsto A'$ , $B \mapsto B'$ , $C \mapsto C'$ in each of these cases.

Side-side-side: $AB = A'B'$ , $BC = B'C'$ and $CA = C'A'$ .
Side-angle-side: $AB = A'B'$ , $AC = A'C'$ and $\angle A = \angle A'$ .
Angle-side-angle: $\angle A = \angle A'$ , $AB = A'B'$ and $\angle B = \angle B'$ .
Angle-angle-side: $\angle A = \angle A'$ , $\angle B = \angle B'$ and $BC = B'C'$ .

Remarks

The side-side-side criterion has its own node. For side-angle-side, the law of cosines gives $BC^2 = AB^2 + AC^2 - 2 \, AB \cdot AC \cos \angle A$ , so $BC = B'C'$ , and case 1 applies. Angle-angle-side reduces to angle-side-angle, because the angle sum gives $\angle C = \angle C'$ . For angle-side-angle, the law of sines then fixes the two other sides.

Two other patterns are not criteria. Side-side-angle fails: $AB = 2$ , $BC = \sqrt{2}$ and $\angle A = 30^\circ$ fit two triangles. By the law of sines $\sin \angle C = \frac{1}{\sqrt{2}}$ , so $\angle C = 45^\circ$ or $\angle C = 135^\circ$ . Angle-angle-angle fails: triangles with equal angles are similar, but they can have different sizes.

Going deeper

For a student

In Euclid's Elements these are early propositions of Book I, and in Hilbert's axioms a form of side-angle-side is taken as an axiom. The criteria are the main tool of synthetic proofs: one shows two segments or angles are equal by fitting them into congruent triangles. Over the reals, each criterion follows from the law of cosines or the law of sines.

Reading path

To understand this page from the ground up, read these 5 pages in order. The foundations come first.

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Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.

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Angle at a point concept
Congruent triangles concept
Euclidean distance concept
Triangle concept
Congruence criteria for triangles statement

Tag	`5NEJPXA6N4`
URI	`m2://5NEJPXA6N4`
Kind	statement, theorem
Writing	W0 stub
MSC	51M04, 51F20
Status	This copy of the site has no build data. Run `m2 status 5NEJPXA6N4`.
About	Congruent triangles
Uses	Congruent triangles, Triangle, Angle at a point, Euclidean distance