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When you have a random triangle $\triangle ABC$, what exactly is the difference between $\angle ABC = 90^o$ and $\angle B = 90^o$? In which cases is it the same, in which cases is it different? What implications does $\angle ABC = 90^o$ have that $\angle B = 90^o$ doesn't and so on and so forth.

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2 Answers

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It depends on the context of the problem. Which one below would be $\angle B$? and $\angle A$?

enter image description here

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Oh, that's it. So in $2D$ it doesn't really matter, however in $3D$.. –  ZafarS Nov 1 '12 at 22:18
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no, in fact the figure is meant to be flat, but calling the angle $\angle B$ here would be ambiguous –  Valentin Nov 2 '12 at 17:39
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This is a matter of convention. Some authors/lecturers will use $\angle ABC$, some will use $\angle B$ and some will use both simultaneously, all to mean the same thing.

The notation $\angle ABC$ is more general in a sense: it can be and is used to refer to the angle made by any three points $A,B,C$ with $B$ at the vertex of the angle. The actual triangle $\triangle ABC$ might not be of immediate relevance. But in my experience, $\angle B$ is typically only used in reference to the interior angle of a triangle, at the vertex $B$ of that triangle. (It might also be used to refer to any angle with vertex $B$, once there is no confusion about which angle is being referred to - as I say, it's a matter of convention and the author's/lecturer's taste.)

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So basically, same angle, different name? –  ZafarS Nov 1 '12 at 22:06
    
@ZafarS: Yes${}$ –  wj32 Nov 1 '12 at 22:08
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Yes. But $\angle ABC$ leaves no room for doubt. –  user12477 Nov 1 '12 at 22:16
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