Question

which must be true in order for the relationship δzyx ~ δwvu to be correct?
○ \overline{zy} || \overline{wv}
○ ∠z ≅ ∠y and ∠w ≅ ∠v
○ \overline{zy} ≅ \overline{yx} and \overline{wv} ≅ \overline{vu}
○ ∠z ≅ ∠w and ∠x ≅ ∠u

Explanation:

Step1: Recall Similar Triangles Criterion

For two triangles to be similar ($\sim$), their corresponding angles must be congruent (AA, SAS, SSS similarity). Here, $\triangle ZYX \sim \triangle WVU$ means $\angle Z \cong \angle W$, $\angle Y \cong \angle V$, $\angle X \cong \angle U$ (corresponding angles).

Step2: Analyze Each Option

• Option 1: $\overline{ZY} \parallel \overline{WV}$ – Parallel sides don't directly imply angle congruence for similarity here. Eliminate.
• Option 2: $\angle Z \cong \angle Y$ and $\angle W \cong \angle V$ – These are not corresponding angles. Eliminate.
• Option 3: $\overline{ZY} \cong \overline{YX}$ and $\overline{WV} \cong \overline{VU}$ – Side congruence (not similarity, which needs proportional sides, not congruent). Eliminate.
• Option 4: $\angle Z \cong \angle W$ and $\angle X \cong \angle U$ – These are corresponding angles. By AA similarity, if two angles are congruent, triangles are similar. This matches the similarity condition.

Answer:

D. $\angle Z \cong \angle W$ and $\angle X \cong \angle U$ (assuming D is the identifier for this option; if the options were labeled as per the order, the last option with $\angle Z \cong \angle W$ and $\angle X \cong \angle U$)