Lines on algebraic varieties

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A variety X is covered by lines if there exist a finite number of lines contained in X passing through each general point. I prove two theorems. Theorem 1:Let Xⁿ⊂ Pᵐ be a variety covered by lines. Then there are at most n! lines passing through a general point of X. Theorem 2:Let Xⁿⁿ⁺¹ be a hypersurface and let x∈ X be a general point. If the set of lines having contact to order k with X at x is of dimension greater than expected, then the lines having contact to order k are actually contained in X.