Exterior Angle Property of a Triangle

An exterior angle of a triangle is formed when any side of the triangle is extended beyond a vertex. The angle formed outside the triangle at that vertex is called the exterior angle.

      A
     /\
    /  \
   B----C---->
       (exterior angle at C)

Every vertex of a triangle can produce two exterior angles, but we usually consider the one that forms a linear pair with the interior angle.

The exterior angle property states that:

An exterior angle of a triangle is equal to the sum of the two interior opposite angles.

For triangle \( \triangle ABC \) with an exterior angle at vertex \( C \):

\( \text{Exterior Angle} = \angle A + \angle B \)

      A
     /\
    /  \
   B----C---->
   | interior angles A & B add up to the exterior angle

The interior angle at the vertex and its exterior angle always form a linear pair, so they add up to \(180^\circ\). From the angle sum property of a triangle:

\( \angle A + \angle B + \angle C = 180^\circ \)

Since the exterior angle = \(180^\circ - \angle C \):

\( 180^\circ - \angle C = \angle A + \angle B \)

Thus, the exterior angle equals the sum of the interior opposite angles.

If an exterior angle is formed at vertex \( C \), then:

• Exterior angle at C = \( \angle A + \angle B \)
• \( \angle C \) is the interior adjacent angle
• The two angles not touching vertex \( C \) are called interior opposite angles

This idea is extremely useful in solving angle-based geometry questions.

Suppose in triangle \( ABC \):

• \( \angle A = 45^\circ \)
• \( \angle B = 55^\circ \)

Then the exterior angle at vertex \( C \) is:

\( \angle A + \angle B = 45^\circ + 55^\circ = 100^\circ \)

This property helps find missing angles even when the triangle is not drawn to scale.

Exterior angles appear in many real-life shapes and structures. When folding paper, designing roof slants, or constructing support frames, the relationship between the outer opening and inner angles naturally follows the exterior angle property.

This predictable relationship makes triangular structures stable and easy to analyze.