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There are countless. For instance, take a look at any bridge. If the angles are not congruent, then BOOM, the whole thing will come down. It will exert to much force on a certain part of the bridge. CivilEngineers need to make sure that the angles are congruent, even to 0.01 degrees. That's just one example.
Shooting pool (billiards).
A real life example would be the two angles on the sides of the Leaning Tower of Pisa.
This is probably a late answer and I have this on my homework but I said...the Hollywood lights the can be both the same length of light going out...idk I think that is write :[
a real life example of an octagon is a stop sign.