Question

which must be true in order for the relationship
\\( \delta z y x \sim \delta w v u \\) to be correct?
\\( \overline{z y} \\| \overline{w v} \\)
\\( \angle z \cong \angle y \\) and \\( \angle w \cong \angle v \\)
\\( \overline{z y} \cong \overline{y x} \\) and \\( \overline{w v} \cong \overline{v u} \\)
\\( \angle z \cong \angle w \\) and \\( \angle x \cong \angle u \\)

Explanation:

Step1: Recall the AA (Angle - Angle) similarity criterion

For two triangles $\triangle ZYX$ and $\triangle WVU$ to be similar ($\triangle ZYX\sim\triangle WVU$), we need two pairs of corresponding angles to be congruent.

Step2: Analyze each option

• Option 1: $\overline{ZY}\parallel\overline{WV}$ does not directly give us the angle - angle condition for similarity.
• Option 2: $\angle Z\cong\angle Y$ and $\angle W\cong\angle V$ is not the correct correspondence of angles for $\triangle ZYX\sim\triangle WVU$.
• Option 3: $\overline{ZY}\cong\overline{YX}$ and $\overline{WV}\cong\overline{VU}$ gives side - side congruence (which is for congruent triangles, not similar triangles in general).
• Option 4: If $\angle Z\cong\angle W$ and $\angle X\cong\angle U$, then by the AA (Angle - Angle) similarity criterion, $\triangle ZYX\sim\triangle WVU$. Because in $\triangle ZYX$ and $\triangle WVU$, if two angles of one triangle are congruent to two angles of another triangle, the triangles are similar.

Answer:

$\angle Z\cong\angle W$ and $\angle X\cong\angle U$