To determine which criterion proves that triangles ABC and A'B'C' are similar, let's analyze the given options:
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OSSS (One Side-Side-Side) - This criterion requires that the corresponding sides of the triangles are proportional. Since we have two side lengths given, we could use this criterion if we can confirm that their corresponding sides are proportional, but we do not have enough information yet.
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OSAS (One Side-Angle-Side) - This criterion can be used when two sides of one triangle are proportional to two sides of another triangle, and the included angle is equal. In this case, we have a 90-degree angle in both triangles, hence OSAS potentially applies.
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OAA (Angle-Angle-Angle) - This criterion states that if two angles of one triangle are equal to two angles of another triangle, then the triangles are similar. Since both triangles have a 90-degree angle, if we can establish that another pair of angles in each triangle are equal, then they would be similar.
Given that both triangles feature a right angle (90 degrees), and if we can demonstrate either another equal angle between them or that two sides are proportional, we can confirm similarity. However, with the options provided, the most straightforward answer is OAA since having two angles (one being the right angle) would suffice to establish similarity.
Therefore, the correct choice that proves the triangles are similar is OAA because all angles of similar triangles are similar as well.
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