Alternate Exterior Angles

Author

Eric W. Weisstein
August 9, 2026

This notebook downloaded from https://mathworld.wolfram.com/notebooks/Trigonometry/AlternateExteriorAngles.nb.

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

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

Illustration

The highlighted congruent angles lie outside the two parallel lines and on opposite sides of the transversal.

Module[{angleColor, lower, mark, phi, upper},
	phi = Pi/3;
	upper = {Cot[phi], 1};
	lower = -upper;
	angleColor = RGBColor[0.15, 0.45, 0.8];
	mark[center_, angles_] := {
		Directive[angleColor, Opacity[0.2]], Disk[center, 0.38, angles],
		Directive[angleColor, AbsoluteThickness[2.5]], Circle[center, 0.38, angles]
	};
	Graphics[{
		mark[upper, {0, phi}],
		mark[lower, {Pi, Pi + phi}],
		AbsoluteThickness[2],
		Line[{{-2, 1}, {2, 1}}],
		Line[{{-2, -1}, {2, -1}}],
		Line[{-2 {Cos[phi], Sin[phi]}, 2 {Cos[phi], Sin[phi]}}],
		Text[Style[Subscript["l", 1], 16, FontFamily -> "Times", FontSlant -> "Italic"], {2.16, 1}],
		Text[Style[Subscript["l", 2], 16, FontFamily -> "Times", FontSlant -> "Italic"], {2.16, -1}],
		Text[Style["t", 16, FontFamily -> "Times", FontSlant -> "Italic"], {1.15, 1.62}],
		Text[Style["α", 18, FontFamily -> "Times", FontSlant -> "Italic"], upper + 0.58 {Cos[phi/2], Sin[phi/2]}],
		Text[Style["α", 18, FontFamily -> "Times", FontSlant -> "Italic"], lower + 0.58 {Cos[Pi + phi/2], Sin[Pi + phi/2]}]
	}]
]