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In this paper, claims by Lemmens and Seidel in 1973 about equiangular sets oflines with angle $1/5$ are proved by carefully analyzing pillar decompositions,with the aid of the uniqueness of two-graphs on $276$ vertices. The NeumannTheorem is generalized in the sense that if there are more than $2r-2$equiangular lines in $\mathbb{R}^r$, then the angle is quite restricted.Together with techniques on finding saturated equiangular sets, we determinethe maximum size of equiangular sets "exactly" in an $r$-dimensional Euclideanspace for $r = 8$, $9$, and $10$.

Equiangular lines and the Lemmens-Seidel conjecture