Exterior Angle of a Triangle
GeometryAn exterior angle of a triangle is formed by one side and the extension of an adjacent side, and it equals the sum of the two non-adjacent interior angles.
Formula
\text{exterior angle} = \text{sum of two non-adjacent interior angles}
Definition
An exterior angle of a triangle is formed by extending one side of the triangle past a vertex. It is the angle between the extended side and the adjacent side of the triangle.
Example
If a triangle has angles $50^\circ$, $60^\circ$, and $70^\circ$, extending the side past the $50^\circ$ vertex creates an exterior angle. That exterior angle $= 60 + 70 = 130^\circ$ (the two angles it is "across" from).
Key Insight
An exterior angle of a triangle is always bigger than either of the two opposite interior angles. This "Exterior Angle Inequality" is used to prove important results about triangles.
Definition
An exterior angle of a triangle is formed by one side and the extension of the other side at a vertex. Exterior Angle Theorem: an exterior angle of a triangle equals the sum of the two non-adjacent (remote) interior angles. It also equals $180^\circ$ minus the adjacent interior angle (since they form a linear pair).
Example
Triangle with angles $A = 45$, $B = 65$, $C = 70$. The exterior angle at $C$: extends side $BC$ past $C$. Exterior angle $= A + B = 45 + 65 = 110^\circ$. Check: $180 - 70 = 110$. Both methods agree.
Key Insight
The Exterior Angle Theorem is a powerful shortcut: to find an exterior angle without measuring, just add the two non-adjacent interior angles. This avoids the two-step subtraction from $180^\circ$ and is used frequently in geometry proofs.
Definition
The exterior angle theorem (Euclid I.32, often) states that an exterior angle of a triangle equals the sum of the two remote interior angles. Proof: if angle $C$ is the interior angle and angle $D$ is the exterior angle at $C$, then $A + B + C = 180$ (angle sum) and $C + D = 180$ (linear pair), so $D = A + B$. This also implies $D > A$ and $D > B$ (Exterior Angle Inequality, Euclid I.16, which holds even in non-Euclidean geometry).
Example
The exterior angle inequality ($D > A$ and $D > B$) is used to prove that the longest side of a triangle is opposite the largest angle, and that all exterior angles of a convex polygon sum to $360^\circ$.
Key Insight
The exterior angle inequality (I.16) is more fundamental than the equality (I.32): it can be proved without the parallel postulate, while the equality requires it. This distinction marks I.16 as a theorem of "neutral" geometry (valid in both Euclidean and hyperbolic geometry), highlighting how deep angle-sum relationships are tied to the parallel postulate.