The question is trying to find the reasons for which triangles XWZ and XZY are similar. We are given that XZ is perpendicular to the line WY. This implies that the angle XZY and XZW have a measure of 90°. In the statement, we are told to complete the reasoning so we can use the SAS similarity postulate to prove they are similar. The SAS similarity postulate says that if in triangles ABC and DEF the angles A and D have the same measure and
[tex]\frac{AB}{DE}\text{ = }\frac{AC}{DF}[/tex]Then the triangles are similar.
Using this, we should apply it to our context. NOte that in the equation we are using the lines that start at the point whose angles are the same. In the statement, A and D are equal, so we use the line AB and the line DE.
In our case, the angles that are the same are the angles located near the point Z. So, in here we want to compare the line XZ from the left side triangle and the line XZ of the right side triangle.
Then, the first fraction would be
[tex]\frac{12}{12}=\text{ }\frac{ZX}{ZX}[/tex]Now, we want to compare the line ZW and the line ZY . So the other fraction would be
[tex]\frac{ZW}{ZY\text{ }}=\frac{16}{9}[/tex]So finally, the comparison we do is
[tex]\frac{12}{12}=\frac{16}{9}[/tex]