Question

determine if triangle u v w and triangle x y z are or are not similar, and, if they are, state how you know. (note that figures are not necessarily drawn to scale.)

Explanation:

Step1: Find the third - angle of triangle $UVW$

The sum of angles in a triangle is $180^{\circ}$. Let the third - angle of $\triangle UVW$ be $\angle V$. Then $\angle V=180^{\circ}-(66^{\circ} + 54^{\circ})=60^{\circ}$.

Step2: Check the angle - angle (AA) similarity criterion

In $\triangle UVW$, the angles are $54^{\circ},60^{\circ},66^{\circ}$. In $\triangle XYZ$, one angle is $66^{\circ}$. If we assume the angles of $\triangle XYZ$ are $\angle Z = 66^{\circ}$, and we find the ratios of the sides.
Let's check the ratios of corresponding sides. $\frac{19}{57}=\frac{1}{3}$, $\frac{22}{66}=\frac{1}{3}$, $\frac{20}{60}=\frac{1}{3}$.
Since the ratios of the corresponding sides of $\triangle UVW$ and $\triangle XYZ$ are equal ($\frac{UV}{XY}=\frac{VW}{YZ}=\frac{WU}{ZX}=\frac{1}{3}$), by the Side - Side - Side (SSS) similarity criterion, the two triangles are similar.

Answer:

The triangles $\triangle UVW$ and $\triangle XYZ$ are similar by the SSS similarity criterion.