# Two congruent triangles

Each triangle has 6 basic determinants: 3 line segments and 3 angles. Is it possible to have two triangles, which have 5 of the basic determinants in common, but are not congruent?

• By "line segment" do you mean the length of a side? – A. I. Breveleri Dec 2 '19 at 5:44
• How is this not just a basic math problem that is frequently flagged and closed around here? – Thomas Markov Dec 2 '19 at 16:01
• @ThomasMarkov Because it takes some creativity to come up with the solution. – Mike Earnest Dec 9 '19 at 1:02

Any triangle with sides $$a,ak,ak^2$$ with $$a+ak\gt ak^2$$ is similar to a triangle with sides $$ak,ak^2, ak^3$$. An interesting thing about this is the value of $$k$$. We have $$k^2-k-1\lt0$$, so $$k=\frac{1\pm\sqrt{5}}{2}$$ (see the Golden ratio) gives the roots, $$k$$ must be positive to avoid negative lengths, and so $$0\lt k \lt\frac{1+\sqrt{5}}{2}$$.