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  "body": "\n## Statement\n\nLet $ABC$ and $A'B'C'$ be [[triangles|0CTNFGYMM2]] in the Euclidean plane.\nWrite $XY$ for the [[distance|0NPPGHQNXQ]] between points $X$ and $Y$, and $\\angle A$ for the [[angle|0GJAJC6TNZ]] $\\angle BAC$, and so on.\nThe triangles are [[congruent|093EDENVA1]] under $A \\mapsto A'$, $B \\mapsto B'$, $C \\mapsto C'$ in each of these cases.\n\n1. Side-side-side: $AB = A'B'$, $BC = B'C'$ and $CA = C'A'$.\n2. Side-angle-side: $AB = A'B'$, $AC = A'C'$ and $\\angle A = \\angle A'$.\n3. Angle-side-angle: $\\angle A = \\angle A'$, $AB = A'B'$ and $\\angle B = \\angle B'$.\n4. Angle-angle-side: $\\angle A = \\angle A'$, $\\angle B = \\angle B'$ and $BC = B'C'$.\n\n## Remarks\n\nThe [[side-side-side criterion|0BRBYRK6PA]] has its own node.\nFor side-angle-side, the [[law of cosines|060ZVZBV3E]] gives $BC^2 = AB^2 + AC^2 - 2 \\, AB \\cdot AC \\cos \\angle A$, so $BC = B'C'$, and case 1 applies.\nAngle-angle-side reduces to angle-side-angle, because the [[angle sum|05MDKQBEAF]] gives $\\angle C = \\angle C'$.\nFor angle-side-angle, the law of sines then fixes the two other sides.\n\nTwo other patterns are not criteria.\nSide-side-angle fails: $AB = 2$, $BC = \\sqrt{2}$ and $\\angle A = 30^\\circ$ fit two triangles.\nBy the law of sines $\\sin \\angle C = \\frac{1}{\\sqrt{2}}$, so $\\angle C = 45^\\circ$ or $\\angle C = 135^\\circ$.\nAngle-angle-angle fails: triangles with equal angles are [[similar|0YF6QXXWB5]], but they can have different sizes.\n\n## Explanations\n\n### L1 human\n\nThese rules say how little you need to know about a triangle to pin it down.\nThree sides are enough, and so are two sides with the angle between them, or one side with two angles.\nEach rule says that two triangles that agree in these pieces are copies of each other.\n\n### L2 human\n\nIn 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.\nThe criteria are the main tool of synthetic proofs: one shows two segments or angles are equal by fitting them into congruent triangles.\nOver the reals, each criterion follows from the law of cosines or the law of sines.\n",
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