Similarity Criteria for Triangles

If two triangles have all three angles equal, the triangles must be scaled copies of each other. Angles control shape. Fix the three angles and you've fixed exactly how the sides lean against each other, so the only freedom left is size. That's why similar triangles always have proportional sides: same shape, different scale.

You don't need to check all three angles. In any triangle the angles sum to 180 degrees, so once two pairs of angles match, the third pair is forced to match too. This is the angle-angle shortcut for proving similarity. Once you know two triangles are similar, every pair of corresponding sides shares the same ratio. That means if you know one side in each triangle, you can find the scale factor, then multiply any other side by that factor to get its match in the other triangle.

This sits right before the two special right triangles and the trig ratios. Those tools work only because right triangles with the same acute angle are automatically similar, so the side ratios never change no matter the size. Sine, cosine, and tangent are really just the side ratios of similarity carried into right triangles. Congruence is the stricter cousin of similarity: congruent triangles are similar with scale factor exactly 1, same size and same shape.