We have some angle, x°, and its supplement, 180°-x°.
When we divide these two by 7 and 13 respectively (so x°/7 and (180°-x°)/13), they'll both essentially equal to each other...
...So let's just do exactly that:
x/7 = (180-x)/13
13x/7 = 180-x (Multiplied both sides by 13)
13x = 1260-7x (Multiplied both sides by 7)
20x = 1260 (Added both sides by 7x)
x = 63° (Divided both sides by 20)