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1. Given any line, there are at least two points on the line. Call them A and B.

2. Given any line, there exists at least one point in the plane that is not on the line. Call that point C.

3. Given any two points (A and C and, then B and C) there exists a straight line joining them.

The point C is not on AB so AB and AC are distinct. Similarly, AB and BC are distinct so that there are three lines that meet, in pairs, at three vertices - and that is a triangle.

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Q: Prove the existence of triangles

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You can't use AAA to prove two triangles congruent because triangles can have the same measures of all its angles but be bigger or smaller, AAA could probably be used to prove two triangles are similar not congruent.

You can prove that to triangles are congruent with SSS, then use CPCTC to prove that two corresponding angles of those triangles are congruent.

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You could prove two triangles are congruent by measuring each side of both triangles, and all three angles of each triangle. If the lengths of the sides are the same, and so are the angles, then the triangles are congruent... if not, then the triangles are not congruent. If the triangles have the exact same size and shape then they are congruent.

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They need no be. You cannot prove an assertion that need not be satisfied.

The colours of their sides.

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