There are many relationships: possibly the simplest one is the Sine rule.
If, in triangle ABC, sides AB, BC and CA are denoted, respectively, by c, a and b [ie each side is denoted by the lower case letter of the upper case letter representing the angle opposite] then
sin(A)/a = sin(B)/b = sin(C)/c
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if any two angles are similar the triangle will be similar
They are of the same lengths
There is no relationship between slope and the theorem, however the theorem does deal with the relationship between angles and sides of a triangle.
The relationship between just the sides is that the sum of any two of them must be greater than the third. Any other relationship involves one (or more) angles.
Relationship between the lengths and the measures of angles are related to theorems like the opposite side of the largest angle is the largest side two equal angles oppositee sides are also equal