Right triangles are congruent if both the hypotenuse and one leg are the same length. These triangles are congruent by HL, or hypotenuse-leg.

Two right triangles can have all the same angles and not be congruent, merely scaled larger or smaller. If all the side lengths are multiplied by the same number, the angles will remain unchanged, but the triangles will not be congruent.

To determine the answer choice that does not lead to congruence, we should simply use process of elimination.

If , then subtracting tells us that .; therefore . Given the fact that reflexively  and that both  and  are both right angles and thus congruent, we can establish congruence by way of Side-Angle-Side.

Similarly, if , then , and given the other information we determined with our last choice, we can establish conguence by way of Hypotenuse-Leg.

If , given what we already know we can establish congruence by Angle-Angle-Side

Finally, if  is an angle bisector, then our two halves are congruent. . Given what we know, we can establish congruence by Angle-Side-Angle

The only remaining choice is the case where . This does not tell us how the two parts of this angle are related, we lack enough information for congruence.

Since we know that , we know that  is also a right angle and is thus congruent to .

We are given that .  Furthermore, since  and  are vertical angles, they are also congruent.  

Therefore, we have enough evidence to conclude congruence by Angle-Side-Angle.  Vertex  matches up with , vertex  matches up with , and  matches up to . Thus, our congruence statement should look the following

We know that congruent triangles have equal corresponding angles and equal corresponding sides.  We are given that the corresponding sides are equal and are in the ratio of .  A triangle whose sides are in this ratio is a , where the shortest side lies opposite the  angle, the longest side is the hypotenuse and lies opposite the right angle, and the third side lies opposite the  angle. (Remember .) So we know the corresponding angles are equal. Therefore, the triangles are congruent.

We know that congruent triangles have equal corresponding angles and equal corresponding sides.  We are given that the corresponding sides are equal and are in the ratio of .

Simplify the ratio by dividing by 

Thus, the corresponding sides are in the ratio  and we know both triangles are  triangles. Since the corresponding angles and the corresponding sides are equal, the triangles are congruent.

We know that congruent triangles have equal corresponding angles and equal corresponding sides.  We are given that the corresponding sides are equal, and the measures of two angles.  We know that  and  because the sum of the angles of a triangle must equal . So the corresponding angles are also equal.  Therefore, the triangles are congruent.

We know that congruent triangles have equal corresponding angles and equal corresponding sides.  We are given that the corresponding sides are equal, and the measures of two angles.  We know that  and  because the sum of the angles of a triangle must equal . So the corresponding angles are also equal.  Therefore, the triangles are congruent.

We know that congruent triangles have equal corresponding angles and equal corresponding sides.  We are given that the corresponding sides are equal and are in the ratio of .  A triangle whose sides are in this ratio is a , where the shorter sides lies opposite the  angles, and the longer side is the hypotenuse and lies opposite the right angle. So we know the corresponding angles are equal. Therefore, the triangles are congruent.

We know that congruent triangles have equal corresponding angles and equal corresponding sides.  We are given that the corresponding sides are equal and are in the ratio of .

Simplify the ratio by dividing by 

Thus, the corresponding sides are in the ratio  and we know both triangles are  triangles. Since the corresponding angles and the corresponding sides are equal, the triangles are congruent.