If two triangles have a side that are of equal length, and also have an equal angle, are those triangles similar?

I know that two triangles are similar if: there are two corresponding congruent angles (AA), if the corresponding sides are of equal proportion (SSS), or if they have two corresponding sides of equal proportion and a corresponding congruent angle in between (SAS)

However, while reading my physics textbook, they throw around the similar triangle argument everywhere. In some instances it looks as if they are claiming that two triangles are similar if you can find an equal angle and an equal side length.

Are there any tricks you know of for determining similar triangles besides the AA, SSS, and SAS?


No, not necessarily, if the equal angles are opposite the equal sides. Take a circle, draw a chord and use it as the base of a triangle. For the third vertex use any point on the arc that goes with the chosen chord. All such triangles have the same base (the chord) and the same angle opposite the base, but they are not similar.

No, not necessarily, even if the equal angles "touch", as you say, the respective equal sides, Say take two triangles each with base $1$, one of them with angles $30^o$ at "the left" and say $50^o$ "at the right (and $100^o$ at "the top"), and for the other triangle $30^o$ at "the left" and say $40^o$ "at the right (and $110^o$ at "the top").

  • $\begingroup$ So if the equal angles "touch" their respective equal sides, the triangles are similar? $\endgroup$ – DWade64 Nov 17 '14 at 23:53
  • 1
    $\begingroup$ no, even if they "touch", I edited my answer (for goodness sake, stay away from physics books !!! :) oops I meant triangle, not rectangle $\endgroup$ – Mirko Nov 18 '14 at 0:00

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