Exterior Angle Theorem Formula

The Formula

\angle_{\text{exterior}} = \angle_{\text{remote}_1} + \angle_{\text{remote}_2}

When to use: Imagine standing at one corner of a triangular park and looking along one side. The exterior angle is how far you'd turn to look back along the other side. That turn combines the 'bends' at the other two corners—it equals their angles added together.

Quick Example

If two remote interior angles are 40° and 75°: \text{exterior angle} = 40° + 75° = 115°

Notation

An exterior angle is formed by one side of the triangle and the extension of an adjacent side

What This Formula Means

An exterior angle of a triangle equals the sum of the two non-adjacent (remote) interior angles.

Imagine standing at one corner of a triangular park and looking along one side. The exterior angle is how far you'd turn to look back along the other side. That turn combines the 'bends' at the other two corners—it equals their angles added together.

Formal View

For \triangle ABC with exterior angle \angle ACD (extending side BC past C): m(\angle ACD) = m(\angle A) + m(\angle B); equivalently m(\angle ACD) = \pi - m(\angle ACB)

Worked Examples

Example 1

easy
In a triangle, two interior angles are 65° and 48°. An exterior angle is formed at the third vertex. Find the exterior angle.

Solution

  1. 1
    Step 1: The Exterior Angle Theorem states: an exterior angle of a triangle equals the sum of the two non-adjacent (remote) interior angles.
  2. 2
    Step 2: The two remote interior angles are 65° and 48°.
  3. 3
    Step 3: Exterior angle = 65° + 48° = 113°.

Answer

The exterior angle is 113°.
The Exterior Angle Theorem provides a shortcut: instead of finding the third interior angle first (180° - 65° - 48° = 67°) and then its supplement (180° - 67° = 113°), you directly add the two remote interior angles. Both methods give the same answer because they both reduce to the same algebra.

Example 2

medium
In \triangle ABC, the exterior angle at C is 130°. If \angle A = 5x° and \angle B = 3x + 2°, find the value of x and both interior angles.

Common Mistakes

  • Using the adjacent interior angle instead of the two remote interior angles
  • Confusing exterior angles with reflex angles
  • Forgetting that each vertex has two equal exterior angles (one on each side)

Why This Formula Matters

A powerful shortcut in proofs and problem-solving—avoids needing to find all three interior angles.

Frequently Asked Questions

What is the Exterior Angle Theorem formula?

An exterior angle of a triangle equals the sum of the two non-adjacent (remote) interior angles.

How do you use the Exterior Angle Theorem formula?

Imagine standing at one corner of a triangular park and looking along one side. The exterior angle is how far you'd turn to look back along the other side. That turn combines the 'bends' at the other two corners—it equals their angles added together.

What do the symbols mean in the Exterior Angle Theorem formula?

An exterior angle is formed by one side of the triangle and the extension of an adjacent side

Why is the Exterior Angle Theorem formula important in Math?

A powerful shortcut in proofs and problem-solving—avoids needing to find all three interior angles.

What do students get wrong about Exterior Angle Theorem?

The exterior angle is supplementary to its adjacent interior angle (\text{exterior} + \text{adjacent interior} = 180°), which is how this theorem follows from the angle sum property.

What should I learn before the Exterior Angle Theorem formula?

Before studying the Exterior Angle Theorem formula, you should understand: triangle angle sum, angles.

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