Solution to Challenge 2

Method
1. The angles around a vertex add up to 360 degrees.
2. Missing angle = 360 - (45 + 80 + 135) = 100 degrees.
3. In order around the vertex the angles are 45, 80, 135, 100.
4. Kawasaki: positions 1 and 3 give 45 + 135 = 180, positions 2 and 4 give 80 + 100 = 180.

Both sums equal 180, so the angles satisfy Kawasaki's theorem and the vertex can fold flat.

Test
- Mark a point on a sheet and draw four rays at those angles with a protractor.
- Choose 3 mountains and 1 valley (Maekawa: |3 - 1| = 2), crease along all four rays and collapse.

Note: the order matters. In the order 45, 135, 80, 100 the odd positions give 45 + 80 = 125, so that vertex would fail.

Checkpoint: missing angle 100 degrees, and both alternating sums equal 180.