Exterior Angle Theorem

The exterior angle is the angle between a side and a line extended from the next side.

The two angles on the inside that are opposite the exterior angle are called remote interior angles.

The Exterior Angle Theorem says:

A triangle's exterior angle d:

  • equals the angles a plus b

and

  • is greater than angle a
  • is greater than angle b
Triangle with interior angles a and b and exterior angle d.

Example:

Triangle with interior angles 35 and 62 degrees and exterior angle 97 degrees.

The exterior angle is 35° + 62° = 97°

And 97° > 35°
And 97° > 62°

Why Is This True?

Because the interior angles of a triangle add to 180°, and angles c and d also add to 180°:

Triangle showing interior angles a, b, c and exterior angle d on a straight line with c.

The interior angles of a triangle add to 180°:a + b + c = 180°Angles c and d make a straight angle (180°):d + c = 180°So d + c equals a + b + c:d + c = a + b + cSubtract c from both sides:d = a + b

Works For Any Triangle's Exterior Angle

Example:

exterior angle theorem 27 and 40 inside triangle, 67 outside

The exterior angle is 40° + 27° = 67°

And 67° > 40°
And 67° > 27°

Example: How big is angle d?

exterior angle theorem 61 inside triangle, d outside

We can't calculate exactly, but we can say:

d° > 61°

The Three Angles Add to 180° Geometry Index