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.

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

The correct answer is none of these. There are several pairs of angles and sides or sides and angles that must be the same in order for two triangles to be congruent.

In our case, we need the acute angle and the hypotenuse to both be equal. No two triangles above have this relationship and therefore no two are congruent.

The congruence of and cannot be proved from the given information alone. Examine the two triangles below:

, , and and are both right angles, so the conditions of the problem are met; however, since the sides are not congruent between triangles - for example, - the triangles are not congruent either.

Right now we can't directly compare these triangles because we do not know all three side lengths. However, we can use Pythagorean Theorem to determine both missing sides. The left triangle is missing the hypotenuse:

The right triangle is missing one of the legs:

subtract 2,304 from both sides

This means that the two triangles both have side lengths 48, 55, 73, so they must be 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.

If we seek to prove that , then , , and correspond to , , and , respectively.

By the Hypotenuse-Leg Theorem (HL), if the hypotenuse and one leg of a triangle are congruent to those of another, the triangles are congruent.

and are both right angles, so and are both right triangles. and are congruent corresponding sides, and moreover, since, each includes the right-angle vertex as an endpoint, they are congruent corresponding legs. and are opposite the right angles, making them congruent corresponding hypotenuses.

The conditions of HL are satisfied, so .

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.