From Twinkle Twinkle Little Star

Each of the '5-stars' and '7-star' is of angle-sum 180 degrees, which make the desired answer to be 900 degrees.

Let the number of points be $n$ , the angles equal to $x_i^\circ$ ( $i = 1,\dots,n$ ), and the number of times the figure turns around $m$ .

In this case, $n = 19$ and $m = 7$ .

If you trace the figure, then at every angle $x^\circ$ you turn $(180-x)^\circ$ . Thus the total angle over which you turn is $180n - \sum_{i=1}^n x_i = 360m.$ Solving this gives $\sum_{i=1}^n x_i = 180n-360m.$ In this case, $\sum_{i=1}^n x_i = 180\times 19-360\times 7 = 180\times(19-2\times 7) = 180\times 5 = \boxed{900}^\circ.$

I don't think I have the same good sights as Chan Lye Lee. Here is my way:

 1
 2
 3
 4
 5
 6
 7
 8
 9
10
11
12
13
14
15
16
17
18
19

A0 =    48
A1 =    48
A2 =    49
A3 =    50
A4 =    51
A5 =    51
A6 =    50
A7 =    50
A8 =    44
A9 =    41
A10 =   50
A11 =   50
A12 =   49
A13 =   50
A14 =   51
A15 =   39
A16 =   40
A17 =   44
A18 =   47

Sum of proximity gave 902. But I know it must be INT(902/ 180)*180, therefore it is equals to 900.

Answer: $\boxed{900}$