外角定理計算機

The Exterior Angle Theorem says: in any triangle, the exterior angle at any vertex equals the sum of the two non-adjacent (remote) interior angles. This is one of the most useful angle theorems in geometry — it lets you compute a third angle from two others without first finding the missing interior angle. This tutorial defines exterior angles, proves the theorem, walks through three worked examples, and shows how the theorem applies to polygons more broadly.

What is an exterior angle?

At any vertex of a triangle (or any polygon), the exterior angle is formed by one side of the figure and the extension of the adjacent side past that vertex.

Visually: at vertex C of triangle ABC, take side BC and extend it past C in a straight line. The angle between this extension and side CA is the exterior angle at C.

An exterior angle is always supplementary to the interior angle at the same vertex (they share a side and the other side is the extension, forming a straight line = 180°):

exterior at C + interior at C = 180°

So an interior angle of 60° has an exterior angle of 120°. An obtuse interior angle of 130° has a (smaller) exterior angle of 50°.

The Exterior Angle Theorem

For triangle ABC, the exterior angle at vertex C (formed by extending BC past C) equals the sum of the two non-adjacent (remote) interior angles ∠A and ∠B:

exterior at C = ∠A + ∠B

Same pattern for the other two vertices: exterior at A = ∠B + ∠C, exterior at B = ∠A + ∠C.

Why is it true?

The proof uses two facts:

1. The three interior angles of any triangle sum to 180°. So ∠A + ∠B + ∠C = 180°.
2. The exterior angle at C is supplementary to interior angle C: exterior + ∠C = 180°.

Combining: exterior at C = 180° − ∠C = (∠A + ∠B + ∠C) − ∠C = ∠A + ∠B. ✓

That's the entire proof — direct algebraic consequence of the 180° sum and the supplementary relation.

Worked example 1 — compute the exterior angle

Triangle has ∠A = 50° and ∠B = 70°. Find the exterior angle at C.

By the theorem: exterior at C = ∠A + ∠B = 50° + 70° = 120°.

Verification: the interior angle at C must be 180° − 120° = 60°. Check: 50° + 70° + 60° = 180°. ✓

Worked example 2 — reverse direction

An exterior angle at C measures 110°. The interior angle at A is 30°. Find ∠B.

By the theorem: exterior at C = ∠A + ∠B → 110 = 30 + ∠B → ∠B = 80°.

Worked example 3 — proving an angle without finding all three interiors

This is where the theorem really shines. In a triangle where two angles are unknown but you know one specific exterior, the theorem may give you a third angle directly without solving for the others.

Example: In a problem you're told the exterior angle at A equals 130° and ∠B = 70°. What is ∠C?

Direct: 130 = ∠B + ∠C → 130 = 70 + ∠C → ∠C = 60°.

You found ∠C in one step. Without the theorem, you'd first compute interior ∠A = 180 − 130 = 50°, then use 50 + 70 + ∠C = 180 to get ∠C = 60° — same answer in two steps.

The exterior angle inequality

A useful corollary of the theorem: each exterior angle of a triangle is greater than either of the two non-adjacent interior angles. (Because it equals their SUM, and both interior angles are positive.)

This was used by Euclid in his Elements to prove several other theorems — most famously, that the longer side of a triangle is opposite the larger angle.

The remote interior angles

The "remote" interior angles (also called "non-adjacent" interior angles) are the two interior angles NOT at the same vertex as the exterior angle. At vertex C's exterior angle, the remote interior angles are ∠A and ∠B (not ∠C).

The "adjacent" interior angle is the one at the same vertex — it's supplementary to the exterior angle, not equal to it.

Exterior angles of any polygon

The theorem about a SINGLE exterior angle is specific to triangles. But a related fact applies to ANY convex polygon: the sum of all exterior angles, one at each vertex (taking one direction of traversal), is always exactly 360°.

For a triangle: three exterior angles summing to 360°. For a quadrilateral: four exterior angles summing to 360°. For an n-gon: n exterior angles summing to 360°.

This is independent of n, which is at first surprising. The geometric meaning: walking around any convex polygon once and turning at each vertex, you make exactly one full turn (360°) by the time you return to start.

Real-world applications

• Surveying. Triangulation calculations use exterior angle relationships to compute distances and bearings without measuring inaccessible interior angles directly.
• Navigation. Triangulating between three landmarks uses both interior and exterior angle theorems.
• Geometric constructions. Many compass-and-straightedge constructions use the exterior angle relationship to bisect or trisect angles.
• Computer graphics. Mesh triangulation and convex hull algorithms rely on the exterior angle sum being 360° to detect when a polygon "closes".

Common mistakes

• Adding all three interior angles to the exterior. The theorem uses only the TWO remote interior angles, not all three. Adding the third gives 180° (the interior sum), not the exterior.
• Confusing exterior with interior at the same vertex. They are supplementary (sum to 180°), not equal. The exterior is the supplement of the interior.
• Applying it to non-triangles. The "exterior = sum of two remotes" theorem is specific to triangles. For polygons with more sides, no single exterior equals a simple sum of remote interiors — the relationships are more complex.
• Treating the extension direction casually. Each vertex of a triangle has TWO possible exterior angles (one on each side of the vertex), but they're vertical angles to each other — both equal. So "the exterior angle" is well-defined and unique in measure.