angle sum identity
It is desired to prove the identities
and
Consider the figure
where we have
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-
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.
Also, everything is Euclidean, and in particular, the interior angles of any triangle sum to .
Call and .From the triangle , we have and , while the degenerateangle , so that
We have, therefore, that the area of the pink parallelogram is . On the other hand, we can rearrange things thus:
In this figure we see an equal pink area, but it is composed of two pieces, of areas and . Adding,we have
which gives us the first.From definitions, it then also follows that , and .Writing