A
pentagram is typically constructed by taking the diagonals of a regular pentagon.
This common pentagram has five angles measured at the vertices, each of which equals 36 degrees. In total all five angles sum to 180 degrees.
Generalize to make an irregular pentagram by taking the diagonals of a convex irregular pentagon.
This irregular pentagram also has five angles measured at the five vertices. Show that the sum of these five angles equals 180 degrees.
Each of the vertices, with angles A1, A2, ..., A5, numbered going around CCW, has a corresponding exterior angle a1, a2, ..., a5, where a1=180-A1, etc.
If you were an ant contiguously walking the lines of the pentagram, turning CCW at each vertex, you would visit vertex 1,3,5,2,4, and arrive back to 1. At each vertex you would turn CCW a1, a3, a5, a2, a4 degrees, finally walking the last line back to vertex 1, and finding yourself pointing in the direction you had started in. Moreover, you would have rotated around your own center exactly twice. Try it. So
a1 + a2 + ... + a5 = 720 degrees
(180-A1) + (180-A2) + ... + (180-A5) = 720
which gives
A1 + A2 + ... + A5 = 180
QED
Edited on February 25, 2021, 8:14 am
soln
