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ice cube
Shooting pool (billiards).
A real life example would be the two angles on the sides of the Leaning Tower of Pisa.
Angles are useful because they allow the natural world to be described accurately. A right angle - 90 degrees - is an example of a particular angle common in modern life, e.g. at the four corners of a table.
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.