Question

determine if triangle qrs and triangle tuv are or are not similar, and, if they are, state how you know. (note that figures are not necessarily drawn to scale.)
sss: three sides proportionate
sss: three sides congruent
sas: two sides proportionate, included angle congruent
sas: two sides + included angle congruent
aa: two angles congruent

Explanation:

Step1: Calculate the ratio of sides

We have two sides of \(\triangle QRS\) as \(19\) and assume the other side (not shown fully in the problem description, but from the ratio perspective) and two sides of \(\triangle TUV\) as \(57\) and \(54\).
The ratio of \(19\) to \(57\) is \(\frac{19}{57}=\frac{1}{3}\).

Step2: Check the included - angle condition

The included angle in \(\triangle QRS\) is \(52^{\circ}\) and the included angle in \(\triangle TUV\) is also \(52^{\circ}\).
Since \(\frac{19}{57}=\frac{1}{3}\) (ratio of two sides) and the included angles (\(52^{\circ}\)) are congruent.

Answer:

Triangles \(QRS\) and \(TUV\) are similar by the SAS (Side - Angle - Side) similarity criterion.