Exterior Angle Theorem Formula

Exterior angle theorem is an exterior angle of a triangle equals the sum of the two non-adjacent (remote) interior angles.

The Formula

exterior angle=remote interior angle1+remote interior angle2\text{exterior angle}=\text{remote interior angle}_1+\text{remote interior angle}_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°40° and 75°75°: exterior angle=40°+75°=115°\text{exterior angle} = 40° + 75° = 115°

Notation

Remote interior angles are the two interior angles not adjacent to the exterior angle.

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 △ABC\triangle ABC with exterior angle ∠ACD\angle ACD (extending side BCBC past CC): m(∠ACD)=m(∠A)+m(∠B)m(\angle ACD) = m(\angle A) + m(\angle B); equivalently m(∠ACD)=π−m(∠ACB)m(\angle ACD) = \pi - m(\angle ACB)

Worked Examples

Example 1

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

Answer

The exterior angle is 113°113°.

First step

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.

Full solution

  1. 2
    Step 2: The two remote interior angles are 65°65° and 48°48°.
  2. 3
    Step 3: Exterior angle =65°+48°=113°= 65° + 48° = 113°.
The Exterior Angle Theorem provides a shortcut: instead of finding the third interior angle first (180°−65°−48°=67°180° - 65° - 48° = 67°) and then its supplement (180°−67°=113°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 △ABC\triangle ABC, the exterior angle at CC is 130°130°. If ∠A=5x°\angle A = 5x° and ∠B=3x+2°\angle B = 3x + 2°, find the value of xx and both interior angles.

Example 3

medium
In △ABC\triangle ABC, the exterior angle at CC measures 4x+10°4x + 10°. The remote interior angles measure 2x+5°2x + 5° and x+25°x + 25°. Find xx.

Common Mistakes

  • Using the adjacent interior angle as a remote angle — remote means not touching the exterior angle.
  • Forgetting the exterior and adjacent interior are supplementary — they form a straight line.
  • Applying the theorem to a non-triangle angle diagram — identify the triangle first.

Why This Formula Matters

This theorem turns triangle angle sums into fast angle chasing. It helps students avoid confusing adjacent linear pairs with remote interior angles. Recognizing it by "Which two interior angles are not touching the exterior angle?" — rather than by familiar numbers — is what lets a student tell it apart from triangle angle sum and linear pair in a mixed problem set.

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?

Remote interior angles are the two interior angles not adjacent to the exterior angle.

Why is the Exterior Angle Theorem formula important in Math?

This theorem turns triangle angle sums into fast angle chasing. It helps students avoid confusing adjacent linear pairs with remote interior angles. Recognizing it by "Which two interior angles are not touching the exterior angle?" — rather than by familiar numbers — is what lets a student tell it apart from triangle angle sum and linear pair in a mixed problem set.

What do students get wrong about Exterior Angle Theorem?

The procedure for exterior angle theorem is the easy part; the trap is using the adjacent interior angle as a remote angle. Asking "Which two interior angles are not touching the exterior angle?" first is what keeps a correct-looking calculation from being attached to the wrong concept.

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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