Angles In A 7-Pointed Star

The sum of all of the angles of the 7 triangles on each side of the heptagon is $7\cdot 180 = 1260$ degrees.

The sum of the exterior angles of the heptagon is $360$ degrees.

The sum of all of the angles of the 7 triangles excluding the angles at the corners is 2 times the sum of the exterior angles (2 sets of exterior angles), which is $2 \cdot 360 = 720$ degrees.

The sum of the angles at the corners is $1260-720=\boxed{540}$ degrees.

= 180(n-4)=3 * 180=540 (you can use this to prove that the least number of angels of a star is five)

Sum of the 7 interior angle of the heptagon = (7-2) 180 = 900 Value of one interior angle = 900/7 = 128.6 Value of one exterior angle of the heptagon = 180 - 128.6 = 51.4 Corner angle = 180 - 51.4 2 = 77.14 Sum of 7 corner angle = 7*77.14 = 540 degrees

a + c + g +e = 360 -x b+d +f=180 +x

Adding the above solves it. x is the interior angle of triangle with vetrex G and triangle with vertex A (oppsite angles)

Try to see seven triangles and a heptagon. The sum of the external angles of the seven triangles is 6300 (900 each). The sum of the internal angles of the heptagon is 900. The external angle of the corners (A - G) is nothing more then 360 - the corner internal angle, and the sum of the external angles of the other corners of the seven triangles is nothing more then 360 * 7 + 2 * Sum of the internal angles of the heptagon (900). That makes an equation:

6300 = 360 * 7 - (A + B + C + D + E + F + G) + 360 * 7 + 2 * 900

6300 = 360 * 14 + 1800 - ( A + B + C + D + E + F + G)

(A + B + C + D + E + F + G) = 540

Assume the picture as a 3D. It would be like a triangle(interior angle sum=180) cut through a quadrilateral (sum of interior angles=360). Hence its 540.

formei 7 quadriláteros com 3 ângulos nas letras e uma ângulo no heptágono do interior do desenho. Somei tudo e deu: 900 + 3(A+B+C+D+E+F+G) = 7*360. Logo A+B+C+D+E+F+G= 540 graus

Name the intersection of GB and AC as O..

name all the angles as there corners like $\angle$ GEC = $\angle$ E

now $\angle$ G + $\angle$ E + $\angle$ C + $\angle$ GOC = 360' --------- 1

Now Join BA

$\angle$ B + $\angle$ A + $\angle$ F + $\angle$ D + $\angle$ OAB + $\angle$ OBA = 360' ---------------2

$\angle$ GOC = $\angle$ AOB (v.o.a)

Add eq 1and 2

$\angle$ A+ $\angle$ B+ $\angle$ C+ $\angle$ D+ $\angle$ E+ $\angle$ F+ $\angle$ G + $\angle$ AOB + $\angle$ OAB + $\angle$ OBA = 360' + 360'

$\angle$ AOB + $\angle$ OAB + $\angle$ OBA = 180' , as they are angles of triangle)

$\angle$ A+ $\angle$ B+ $\angle$ C+ $\angle$ D+ $\angle$ E+ $\angle$ F+ $\angle$ G = 720' - 180'

$\angle$ A+ $\angle$ B+ $\angle$ C+ $\angle$ D+ $\angle$ E+ $\angle$ F+ $\angle$ G = 540'

Since the specific angles are not specified, assume that this is a regular heptagonal star. The angles of each of the equal angles in the triangles will be 360/7. So the angle of the odd angle out is 180 - 2(360/7). We get:

180 -720/7 1260/7 - 720/7 540/7. And since there are seven of these, our answer for the total is 540.

there are 7 triangle let we have to find the angle A,B,C,D,E,F,G Also, there are triangle APQ,BQR ,CRS ,DST , ETU , FUV ,GVP triangle APQ, angle A = 180 - <1-<2 triangle BQR ,angle B = 180 - <2 - <3 triangle CRS , angle C = 180 - <3- <4 triangle CST , angle D = 180 - <4 - <5 triangle ETU , angle E = 180 - <5 - <6 triangle , FUV , angle F = 180 - <6 - <7 triangle , GVP , angle G = 180 - <7 - <1 SO , angle (A+B+C+D +E+F+G) = 1260 - 2(1+2+3+4+5+6+7) -----------------------------------------------(1) = 1260 - 2(180 -<A + 180 - <C + 180 - <F + <5) = 1260 - 2(540) - 2(angle A+C+F -< 5) = 1260 - 1080 - 2(angle A+C+F - <5) ------------------------------------------(2) also, in quadrilateral , ACTF, angle (A+C+<CTF + F) = 360 angle (A+C)+(180-<5)+ F = 360 angle (A+C+F) - < 5 = 180 ----------------------------------------------------(3) putting, Eqn. (3) into (2) , we get , angle ( A+B+C+D+E+F+G) = 540

360+180

corner angle = pi - (pi - adjacent interior angle of inner 7 sided polygon) - (pi - other adjacent interior angle of inner 7 sided polygon).

summation(corner angle) = 2 summation(interior angles) - 7 pi = 15 pi - 7 pi = 3*pi = 540 degrees