how the test of similarity of two triangles confirm that they are similar?

Question

For two triangles to be similar , their corresponding angles should be congruent and corresponding sides should be in proportion . But for given two triangles we don't always need to check whether their all angles are congruent and all corresponding sides are in proportion instead we use some tests of similarity which only use some aspects of the two triangles and on verifying them we can conclude that the two triangles are similar . Examples of such test are A-A test , S-S-S test , and S-A-S test . How these tests imply that the two triangles are similar ? Here S-S-S test means corresponding sides of one triangle are in proportion to the corresponding sides of the other triangle and S-A-S test means two corresponding sides of first triangle are in proportion with the two corresponding sides of the other triangle and the included angle of both the triangles is congruent.