Question

can right triangles with proportional sides have angles that are not congruent and therefore not be similar

Answer 1

Task. Can right triangles with proportional sides have angles that are not congruent and therefore not be similar?

Solution. Recall that two triangles $ABC$ and $A^{\prime}B^{\prime}C^{\prime}$ are called similar if they have proportional sides, i.e.

$\frac{AB}{A^{\prime}B^{\prime}}=\frac{AC}{A^{\prime}C^{\prime}}=\frac{BC}{B^{\prime}C^{\prime}}.$

There is also a theorem from geometry claiming that two similar triangles $ABC$ and $A^{\prime}B^{\prime}C^{\prime}$ have equal corresponding angles, i.e.

$\angle A=\angle A^{\prime},\qquad\angle B=\angle B^{\prime},\qquad\angle C=\angle C^{\prime}.$

In our case we have right triangles with proportinal sides. Then by definition these triangles are similar. It does not matter whenther they are right or not. Hence by theorem above, these triangles have equal corresponding angles.

Thus the answer to the question is ”No”.