Question

1. in the figure below, ∠qts ≅ ∠trs.

(other parts: use the drop - down menus to complete statements about the similar triangles... choose one option from each drop - down... two triangles are similar if... with some drop - down options like the triangles share the same vertex, adjacent angles are congruent and some choose... menus)

Explanation:

Step1: Recall Similar Triangles Criterion

Two triangles are similar if their corresponding angles are congruent (AA, SAS, SSS similarity criteria). Here, we analyze the angle relationships. Given \( \angle QTS \cong \angle TRS \), and \( \angle S \) is common to both \( \triangle QTS \) and \( \triangle TRS \). So, by AA (Angle - Angle) similarity, if two angles of one triangle are congruent to two angles of another triangle, the triangles are similar. The relevant option for the condition of similarity here is "adjacent angles are congruent" (since we can use the common angle and the given congruent angle to establish AA similarity).

Step2: Identify Similar Triangles

Looking at the figure, \( \triangle QTS \) and \( \triangle TRS \) share \( \angle S \), and \( \angle QTS \cong \angle TRS \). So the similar triangles would be \( \triangle QTS \sim \triangle TRS \) (by AA, as two angles are congruent: \( \angle S \) is common, \( \angle QTS \cong \angle TRS \)).

Answer:

For the "Two triangles are similar if" drop - down, the correct option is "adjacent angles are congruent". The similar triangles are \( \triangle QTS \) and \( \triangle TRS \) (depending on the other drop - downs, but for the similarity condition part, the key is the angle congruence condition).