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.

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.

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. 

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 .

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. 

We can find one leg of a right triangle when we have the length of the hypotenuse and the other leg.  Square the hypotenuse and the known leg.  Then, subtract the squared length of the leg from the squared length of the hypotenuse.  Finally, find the square root of the result.

If  is not the base, that makes either  or  the base. If either  or  is the base, the right angle is on the bottom, so  or  respectively will be perpendicular. The height of a triangle is the distance from the base to the highest point, and in a right triangle that will be found by the side adjoining the base at a right angle. So if the base is , then  and vise versa.

We begin with the formula for the area of a triangle.

We further realize that the two legs of the triangle are the base and height; therefore, substituting what we know we get

We can then solve by simply dividing.

We have found our height and thus the second leg of our triangle.