Reiteration of explanation given within image:

Step 1:
Let’s call the intersection of all the straight lines X.
∠MXQ and ∠PXN are opposite angles/ vertical angles, meaning that they are the same. So, ∠MXQ = ∠PXN. (Similarly, ∠MXR = ∠SXN, and ∠RXP = ∠QXS)
You know that a triangle has a total internal angle of 180, regardless of the size and shape. Thus, b + c = f + g. You can think of it this way:
f + g + ∠MXQ = 180 –(1)
b + c + ∠PXN = 180 –(2)
Since ∠MXQ = ∠PXN,
You can rewrite (1) as:
f + g + ∠PXN = 180 –(3)
(3)-(2), f + g + ∠PXN – (b + c + ∠PXN) = 180 - 180
Thus, f + g – b – c = 0
Which gives you f + g = b + c
Step 2:
Draw a line from M to Q. Now, you get a quadrilateral, RMQS. Recall that the total internal angle of a quadrilateral is 360. This means that ∠MRS + ∠RSQ + ∠SQM + ∠QMR = 360.
You already know that ∠MRS = 80. What about the rest? Looking at the diagram will give you:
∠RSQ = d ∠SQM = g + e ∠QMR = a + f
Now, substitute the four angles into our quadrilateral formula:
∠MRS + ∠RSQ + ∠SQM + ∠QMR = 360
80 + d + g + e + a + f = 360
Rearranging, you get:
a + f + g + d + e = 360 - 80 = 280
Since f + g = b + c (derived from step 1),
You can conclude that a + b + c + d + e = 360 - 80 = 280