We have altitude h to side AB and AB=h, i.e. the altitude is congruent to the side it goes to.
That's all kinds of triangles. One way to see them is using two horizontal parallel lines h apart, the bottom one with a base AB=h somewhere on it. Then any C on the top line makes a triangle ABC with altitude h=AB.
Let's go through the choices.
I. ABC could be a right triangle. That's TRUE.
We could have the isoscleles right triangle, C directly above B, so AC is the leg and an altitude, AB=AC and B is the right angle.
II. Angle C cannot be a right angle. That's TRUE.
The biggest angle C can be is when it's over the midpoint of AB, so if AB=2, h=2, and
[tex]AC=\sqrt{2^2+1^2}=\sqrt{5}[/tex]
so
[tex]C_{\textrm{max}} = 2 \arctan(1/2) \approx 53.13^\circ[/tex]
III. Angle C could be less than 45 degrees. That's TRUE.
As long as C stays on our top parallel, we can make it as acute as we like by going farther away from AB.
All true. Hmmm.
