Answer to Question #140952 in Geometry for J

Question

using the exterior angle inequality prove that any right triangle has only one right angle and the other two angles must be acute.

Answer 1

We have a triangle $\triangle ABC$ , where angle $\angle ABC$ is right, and angles $\angle CAB, \angle BCA$ are not right.

If angle $\angle ABC$ is right it equals to 90 degrees ($\angle ABC = 90\degree$) then exterior angle

$\angle ABC$_exterior is equal to $180\degree - 90\degree = 90\degree$ ( $\angle ABC$_exterior$= 90\degree$ ), because any exterior angle $\alpha$_exterior is equal to $180\degree - \alpha$.

During the exterior angle inequality, exterior angle of a triangle is greater than either of the non-adjacent interior angles. So $\angle CAB < \angle ABC$_exterior and $\angle BCA < \angle ABC$_exterior , it means that

$\angle CAB < 90\degree$ and $\angle BCA < 90\degree$. So, both of this angles are acute.