Lucky Star Number 7

Notice that there is a septagon inside the inside the star. We can find the angle of each side simply enough: $(7-2)\cdot180/7=(5\cdot180)/7$ . This is $\approx128.6$ degrees. The triangles formed have 2 equal angles. We can find them easily, as they are supplementary to the angle we found already: $x\approx180-128.6$ . This angle is $\boxed{\approx51.4}$ . Now, finding the last angle is easy: $a\approx180-(51.4\cdot2)\approx77.1$ . So the final answer is $\boxed{77}$ .

The total internal angles of a regular n-gon = 180(n-2) (See more example at Polygon Complex )

Therefore, the angle of each interior angle in a heptacle = 180(7 - 2)/7 = 180(5/7).

Hence, the exterior angle (base angle of the external triangle) = 180(1 - (5/7)) = 180(2/7)

Considering the external triangle, the blue angle = 180 - 2[180(2/7)] = 180(1 - (4/7)) = 180(3/7) ≈ 77 degrees.

(77 degrees in each of the 7 triangles surrounding the 7-gon? Is it merely just a coincidence or purely beauty of mathematics? You decide.)

Note : The total pointing angles of any n-star = 180(n - 4).

That is, the total exterior angles = 2(base angle)(number of triangles) = 2(360/n)(n) = 720 = $4\times 180$ .

Therefore, we have to subtract 4 triangles out from all the $n$ triangles in order to get the total angles pointing outwards.