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An N -tiling of triangle ABC by triangle T is a way of writing ABC as a unionof N triangles congruent to T, overlapping only at their boundaries. Thetriangle T is the "tile'".  The tile may or may not be similar to ABC . This paper is the first ofseveral papers, which together seek a complete characterization of the triples(ABC,N,T) such that ABC can be N -tiled by T . In this paper, we consider thecase in which the tile is similar to ABC, the case in which the tile is a righttriangle, and the case when ABC is equilateral.  We use (only) techniques from linear algebra and elementary field theory, aswell as elementary geometry and trigonometry.  Our results (in this paper) are as follows: When the tile is similar to ABC,we always have "quadratic tilings'" when N is a square. If the tile is similarto ABC and is not a right triangle, then N is a square. If N is a sum of twosquares, N = e^2 + f^2, then a right triangle with legs e and f can be N -tiledby a tile similar to ABC ; these tilings are called "biquadratic". If the tileand ABC are 30-60-90 triangles, then N can also be three times a square. If Tis similar to ABC, these are all the possible triples (ABC, T, N) .  If the tile is a right triangle, of course it can tile a certain isoscelestriangle when N is twice a square, and in some cases when N is six times asquare.  Equilateral triangles can be 3-tiled and 6-tiled and hence they can also be3n^2 and 6n^2 tiled for any n . We also discovered a family of tilings when Nis 3 times a square, which we call the "hexagonal tilings." These tilingsexhaust all the possible triples (ABC, T, N) in case T is a right triangle oris similar to ABC . Other cases are treated in subsequent papers.

Triangle Tiling I: The tile is similar to ABC or has a right angle