Question

in the figure below, $\angle qts \cong \angle trs$.
use the drop - down menus to complete the statements about the similar triangles in the figure.
choose one option from each drop - down menu to answer the question.
two triangles are similar if
choose...
$\triangle qt$ choose...
this
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choose..
two corresponding exterior angles are congruent
two pairs of corresponding angles are congruent
the sum of interior angles are the same.

Explanation:

Step1: Recall Triangle Similarity Criterion

The AA (Angle - Angle) similarity criterion states that two triangles are similar if two pairs of corresponding angles are congruent. Exterior angles are not used to define triangle similarity, and the sum of interior angles of any triangle is always \(180^{\circ}\), so that does not define similarity. So the correct option for the first drop - down (two triangles are similar if) is "two pairs of corresponding angles are congruent".

Step2: Identify Similar Triangles

Looking at the figure, we have \(\angle QTS\cong\angle TRS\) and \(\angle S\) is common to \(\triangle QTS\) and \(\triangle TRS\). So the similar triangles should be \(\triangle QTS\) and \(\triangle TRS\) (assuming the first triangle is \(\triangle QTS\) and the second is \(\triangle TRS\), but from the partial notation \(\triangle QT\) (maybe a typo, should be \(\triangle QTS\)) and the other triangle. Since we have two pairs of congruent angles (\(\angle QTS\cong\angle TRS\) and \(\angle S\cong\angle S\)), by AA similarity, \(\triangle QTS\sim\triangle TRS\).

Answer:

For the first drop - down: two pairs of corresponding angles are congruent.
For the triangle similarity: \(\triangle QTS\sim\triangle TRS\) (assuming the relevant triangles, with the first triangle being \(\triangle QTS\) and the second \(\triangle TRS\) based on the angle markings and common angle).