Inspired by Twinkle Twinkle Little Star

Using Vertical Angles and Triangles - Exterior Angles , we can see:

Using Regular Polygons - Angle Sum , the internal heptagon (outlined in blue) has a sum of angles equal to $5\cdot 180$ degrees. However, each of its angles is in the form $180-a,$ for a total sum of $180\cdot 7 - (a+b+c+d+e+f+g),$ so $a+b+c+d+e+f+g = 2\cdot 180.$

Each of the angles we want to add up is in the form $180-a-b,$ and each variable is included in two triangles. Thus, the entire sum is $180\cdot 7 - 2(a+b+c+d+e+f+g) = 180 \cdot 7 - 2\cdot 2 \cdot 180= 3 \cdot 180 = \boxed{540}.$

Remark: There are many clever ways to approach this problem (e.g., the "pencil rolling" approach), and the result can be generalized. How did you solve it? Post your solution!

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Here is my approach:

It is easy to show that the sum of angles in '5-star' is 180 degrees, then the other two triangles has angle sum 180 degrees each. Hence the answer.

Travel through the diagram from A back to A. It is clear that you make two complete loops around the center, so that you turned $2\cdot 360^\circ$ .

The marked angle at each turn is $180^\circ -\$ the angle over which you turn when traveling around.

Therefore, $2\cdot 360^\circ = 7\cdot \sum_{X = A,\dots,G} (180^\circ - \angle X) = 7\cdot 180^\circ - (A + \dots + G).$ Solve this to find $A + \dots + G = 7\cdot 180 - 2\cdot 360 = 540^\circ$ .