The triangles in the picture are congruent by which method.

Explanation:

Check out the diagram below. I've marked the angles that pair up which are congruent to one another. Note the order is important. The AAS rule is different from ASA. In the diagram, the green segments are not between the red and blue angles.

Note: the blue angles are congruent by the vertical angle theorem.

we have three similar triangles, because each has a right angle and shares an angle.   let's write the angles in order: opposite to short leg, long leg, hypotenuse.

cab similar to bad similar to cbd

or as ratios,

ca: ab: cb = ba: ad: bd = cb: bd: cd

we also know

ac = ad + cd

(ad+cd): ab: cb = ba: ad: bd = cb: bd: cd

(ad+cd)/cb=cb/cd

we have cb=6, ad=5 and seek x=cd.

we reject the negative root and conclude x=4

answer: 4