In the figure, one exterior angle is drawn at each vertex of the triangle. Explain how to find the sum of the measures of these three exterior angles.

Explanation:

Here's one way:

At vertex A, draw the vertical angle to the given angle 4. That is, extend AC, rather than AB to show the exterior angle at that vertex.

Now, collapse triangle to one with zero area, so that vertices A, B, C lie on top of each other. You can do this using a dilation scale factor f such that f→0.

The result is that the exterior angles divide the circular space around point ABC into three parts that sum to 360°. Essentially, we have copied each of the exterior angles to the same vertex.

The sum of the measures of the exterior angles is 360°.

Additional comment

This identical approach can be used to show the sum of exterior angles around any convex polygon is always 360°.

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