SAS Similarity Theorem

Author

Eric W. Weisstein
August 16, 2026

This notebook downloaded from https://mathworld.wolfram.com/notebooks/PlaneGeometry/SASSimilarityTheorem.nb.

For more information, see Eric's MathWorld entry https://mathworld.wolfram.com/SASSimilarityTheorem.html.

©2026 Wolfram Research, Inc. except for portions noted otherwise

Figure

The two triangles have an equal included angle, and the adjacent corresponding side lengths have the common ratio k.

Module[{angleMark, blue, k, labelPositions, labels1, labels2, neutral, red, triangle1, triangle2},
	triangle1 = {{-3., -1.}, {-0.3, -1.}, {-2.1, 1.2}};
	k = 0.65;
	triangle2 = (k RotationTransform[-0.55][# - triangle1[[1]]] + {0.7, -0.6} &) /@ triangle1;
	neutral = LightDarkSwitched[Black, White];
	blue = Blue;
	red = Red;
	labelPositions[triangle_] := (# + 0.2 Normalize[# - Mean[triangle]] &) /@ triangle;
	angleMark[triangle_, radius_] := Module[{a, finish, start, u, v},
		a = triangle[[1]];
		u = triangle[[2]] - a;
		v = triangle[[3]] - a;
		start = Apply[ArcTan, u];
		finish = Apply[ArcTan, v];
		{Directive[red, Opacity[0.18]], Disk[a, radius, {start, finish}],
		 Directive[red, Opacity[1], AbsoluteThickness[2.5]], Circle[a, radius, {start, finish}]}
	];
	labels1 = {"A", "B", "C"};
	labels2 = {"A'", "B'", "C'"};
	Graphics[{
		Directive[neutral, AbsoluteThickness[2]], Line[Append[triangle1, First[triangle1]]],
		Directive[blue, AbsoluteThickness[2.5]], Line[Append[triangle2, First[triangle2]]],
		angleMark[triangle1, 0.48], angleMark[triangle2, 0.31],
		MapThread[Text[Style[#1, 16, neutral, FontFamily -> "Times", FontSlant -> "Italic"], #2] &,
			{labels1, labelPositions[triangle1]}],
		MapThread[Text[Style[#1, 16, blue, FontFamily -> "Times", FontSlant -> "Italic"], #2] &,
			{labels2, labelPositions[triangle2]}],
		Text[Style["θ", 17, red, FontFamily -> "Times"], triangle1[[1]] + {0.58, 0.25}],
		Text[Style["θ", 17, red, FontFamily -> "Times"], triangle2[[1]] + {0.38, 0.02}],
		Text[Style["b", 16, neutral, FontFamily -> "Times", FontSlant -> "Italic"], {-2.68, 0.22}],
		Text[Style["c", 16, neutral, FontFamily -> "Times", FontSlant -> "Italic"], {-1.65, -1.13}],
		Text[Style["kb", 16, blue, FontFamily -> "Times", FontSlant -> "Italic"], {1.19, 0.03}],
		Text[Style["kc", 16, blue, FontFamily -> "Times", FontSlant -> "Italic"], {1.53, -1.22}]
	}]
]