What is A＋B＋C＋D＋E?
A hard-level "Angle" problem.Nowhere does this argument use "because it is regular". It uses only two facts: a triangle's angles total 180°, and any polygon's exterior angles total 360°. So however you distort the points, the sum is always 180°.
Hints
- Each of the five points is coloured differently; we want their total. The shape of the star does not matter — everything that follows holds just as well for a long, distorted star.
- First look at the pentagon inside the star, and nothing else. The star's lines have been faded out so only the pentagon is left. It is the foundation of the whole argument.
- Now keep only the five point-triangles — one stuck on the outside of each side of the pentagon. A triangle's angles total 180°, so five give 5 × 180 = 900°.
- Split up what that 900° is made of: the coloured five point angles (what we want) and the ten orange base angles. So subtract the ten base angles from 900° and the points are left. All that remains is to find those base angles.
- The star is gone; the situation is redrawn large. Blue is the inner pentagon, orange the point-triangle.Look at the thick horizontal line: the point ア, the pentagon vertex and the next vertex all lie on one straight line, because the star's edges are straight.Below that line there are only two angles, filling it completely: the pentagon's interior angle (blue) and the triangle's base angle (orange). A straight line is 180°, so interior + base = 180°.
- So each base angle is exactly an exterior angle of the pentagon — the one that makes 180° with the interior angle. Back to the pentagon alone, with all ten base angles filled in: two at every vertex, one from each neighbouring triangle.
- Now the rule: the exterior angles of any polygon always total 360° — however many sides, however distorted — because walking once round turns you through exactly one full turn. Two per vertex, so the ten base angles total 360 × 2 = 720°.
- Bring the star back. Take the 720° of base angles off the 900° and what is left is the total of the five points: 900 − 720 = 180°. Nowhere did we use that the star is regular — only the triangle sum and the exterior-angle sum. So every star gives 180°.
- Finally, a visual check. If the total really is 180°, cutting out the five points and gathering them at one spot should give exactly a half turn — one straight line, no gaps, no overlaps. And it does. This is a check, not the proof; the proof was the subtraction.
Answer 180°
More "Angle" problems
- How big is angle x?
- How many degrees is angle x?
- How big is angle BIC?
- How many degrees is angle x?
- How many degrees is angle x?
- What is the angle at the centre of the sector?