### One of a continuing series

This popped up on my Facebook time line, posted by somebody else. So I stole it, and here it is: What is the sum of all the blue angles? Post your answer as a comment below.

## Update

A number of people have posted responses, so I am going to supply the answer. See the following diagram:

What is the sum of interior angles of a polygon? The example of a triangle explains. The triangle is **ABC**, defined by its three interior angles. But concentrate on the complementary angles **a** and **b** and **c**. What is the sum of those angles? Consider the line **ab**. Line **bc** branches off from **ab** with a change of direction equal to angle **b**. Follow the path around the triangle, and the total change of direction is 360 degrees. That’s going to be the total of **a** and **b** and **c**. The sum of **a** and **A** is 180° so the sum of all angles is 3 × 180 = 540. 540 – 360 = 180, the sum of interior angles of all triangles.

The method holds true for all polygons. The polygon in this puzzle is unusual in that the path makes two complete turns or 360 × 2 = 720. There are 6 sides and six interior angles, so the sum of the interior angles is 6 × 180 – 720 = 360.

Has to be 360, since tracing a single triangle twice would fit the requirement.

I’ve spent the last few minutes trying to figure out why a triangle’s inner angles don’t sum to 360, since a line rotated to meet each side would rotate a total of 360, but I finally understood it’s because each of those rotations would be the outer angle with the inner being 180 – outer. So the inner angles sum to 3 * 180 – 360 or 180. For N vertices and M complete rotations it would be N * 180 – M * 360, so for the Facebook puzzle that’s 720 – 360 or the same as the intuitive overlaying of two triangles.

(All units degrees but I’m using my phone 🙂 )

180 – interior1 + 180 – interior2 + 180 – interior3 = 540 – sum of interiors

But the sum of the interiors is 180, so the difference is 360.

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