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Aa PostulateThe AA Postulate in Euclidean geometry states that two triangles are similar if they have two corresponding angles congruent. The AA Postulate works because the interior angles of a triangle are always equal to 180°. By knowing two angles, such as 32° and 64° degrees, we know that the next angle is 84°, because 180-(32+64)=84. (Some people refer to this as the AAA Postulate, which is true in all respects, but you truly only need two angles.) We can help to understand the postulate by working in reverse order. We can start with two triangles on grids A and B which are similar, by a 1.5 dilation from A to B. If we line up the two triangles, such as in C, we find that the angle on the origin is congruent with the other one (D). We also know that the pair of sides opposite the origin are parallel. We know this because the pairs of sides around them are similar, stem from the same point, and line up with each other. We can then look at the sides around the parallels as transversals, and therefore the corresponding angles are congruent. Using this reasoning we can tell that similar triangles have congruent angles. Postulates cannot be proven entirely however, as any logical person will tell you, just because a results in b doesn't mean b results in a. (Such as, all squares are rectangles, but not all rectangles are squares.) However, we can look at any set of triangles that have congruent angles, and call the two triangles similar, but there is no way to prove why, postulates are accepted in geometry without question.
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