Def: Two triangles are called similar if and only if one can be scaled into the other.
AAA Theorem: If two triangles have the same angles if and only if they are similar.
How does one prove the Theorem, without relying on the trigonometric functions? I can see how to prove one direction with them (but not without them):
It is easy to say, if the angles are $a,b,c$ , the sides in the first triangle are $A_1,B_1,C_1$, and the sides in the second triangle are $A_2,B_2,C_2$. Then by law of sines,
$$\frac{A_1}{\sin a}=\frac{B_1}{\sin b}=\frac{C_1}{\sin c}$$
$$\frac{A_2}{\sin a}=\frac{B_2}{\sin b}=\frac{C_2}{\sin c}$$.
From which by division we see:
$$\frac{A_1}{A_2}=\frac{B_1}{B_2}=\frac{C_1}{C_2}=k$$
Where $k \in \mathbb{R}$ is some constant depending on the two triangles (ratio of two sides).
And hence $(A_1,B_1,C_1)$ is a multiple of $(A_2,B_2,C_2)$. Then up to some scaling, the sides of the triangles are the same size. So up to some scaling, the triangles are congruent by SSS.
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$\begingroup$Dilate one of the triangles until one of its sides is the same length as the corresponding side of the other triangle. Dilation preserves angle measures, so they still have all their angles equal. It follows from the ASA postulate that the triangles are now congruent (and hence that the original triangles were similar).
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