# AAA similarity Theorem.

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.

• Euclid Book VI, Definition 1 and Proposition 5 are probably what you want. – Chappers Jul 25 '17 at 22:26
• On behalf of @Dan von Bose: AA could be considered a theorem since the third angle is free after knowing two of them. AAA is an axiom to combine with the congruency theorems SAS, ASA, and SSS to prove simiilarity theorems SAS and SSS. It depends. What axioms do you want to start with? – dantopa Dec 22 '18 at 1:50

• Thanks, but what about: The triangles are similar $\implies$ angles are equal. – Ahmed S. Attaalla Jul 25 '17 at 22:44
• Actually I guess: if they triangles are similar then up to some scale $k$ they are congruent , because scaling preserves angles, (scaling by $\frac{1}{k}$) the angles must be the same. – Ahmed S. Attaalla Jul 25 '17 at 22:52