Which must be true in order for the relationship ^zyx~^wvu to be correct

Similar triangles may or may not be congruent.

The relationship that must be true for is  (d) and

From the question, we understand that:

The above means that, both triangles are similar (but not congruent). This means that:

In other words

,   and

Hence, the true relationship is (d) and  

Read more about similar triangles at:

link

Correct option: fourth one: ∠Z = ∠W and ∠X = ∠U

Step-by-step explanation:

There are 3 cases that gives similarity of two triangles:

Side-Angle-Side: They should have 2 sides and 1 angle in common

Side-Side-Side: They should have all 3 sides in commom

Angle-Angle: They should have 2 angles in common (therefore the third angle will also be equal)

In the first option, we have a pair of sides being parallel. That is not enough to prove similarity.

In the second option, two angles of the same triangle are equal, that's not enough, because the angles between each triangle can be different.

In the third option, two side of the same triangle are equal, that's not enough.

In the fourth option, two angles of the triangles are equal to each other, so this is the third case mencioned above, therefore it proves the similarity.

Correct option: fourth one.