Because triangles add up to 180 degrees
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Corresponding angle are used to prove if lines are parallel. If they are congruent then the lines cut by the transferal are parallel.
You can prove that to triangles are congruent with SSS, then use CPCTC to prove that two corresponding angles of those triangles are congruent.
It is important so you can later prove that the figure is congruent using a geometric proof or other method. It is also useful for finding side lengths and the measures of angles.
Just because they look the same does not mean they are. Normally if sides are congruent they are marked by the same indicator. Which is usually a line, number, or letter. Same thing with angles, except angles would not use lines to indicate congruency. If you have any three congruent parts on triangles (the number may change due the shape) then those triangles are congruent.
Before using Corresponding Parts of a Congruent Triangle are Congruent theorem (CPCTC) in a geometric proof, you must first prove that there is a congruent triangles. This method can be used for proving polygons and geometrical triangles.