Two Definitions of Exterior Angle in GRE Geometry

This may come as a surprise, but there is more than one way to classify an exterior angle. You will want to have both of these definitions in mind when dealing with questions on the test since both instances are possible. Don’t worry! It won’t be too difficult to remember.

Type I: At each vertex, extend the segment, and measure the angle of the “turn”.

tdoea_img1 For any convex polygon, this kind of exterior angle is always less than 180° at each vertex.  For this kind of exterior angle, the sum of all the exterior angles for any polygon is 360°.

 

 

 

 

 

Type II: At each vertex, take all of the angle not inside the polygon

tdoea_img2

For any convex polygon, this kind of exterior angle is always greater than 180° at each vertex.  The sum of these is much bigger, and depends on the kind of polygon.

Let n be the number of vertices in a polygon (n = 3 for triangle, n = 4 for a quadrilateral, etc.)

We have to add up all the circles, and subtract the angles inside the polygon.

 

(sum of exterior angles) = n*(360°) – (sum of angles inside the polygon)

=  n*(360°) – (n – 2)*180°

For a triangle (n = 3), sum of Type II exterior angles = 900°

For a quadrilateral (n = 4), sum = 1080°

For a pentagon (n = 5), sum = 1260°

For a hexagon (n = 6), sum = 1440°

 

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