Question

determine if triangle def and triangle ghi are or are not similar, and, if they are, state how you know. (note that figures are not necessarily drawn to scale.)

Explanation:

Step1: Calculate the ratios of corresponding sides

For triangle \(DEF\) and \(GHI\), consider the sides. Let's assume the sides adjacent to the \(41^{\circ}\) angles.
The ratio of the sides: \(\frac{6}{18}=\frac{1}{3}\), \(\frac{9}{27}=\frac{1}{3}\)

Step2: Check the included - angle condition

The included angle in \(\triangle DEF\) (the angle between sides \(DF = 6\) and \(DE=9\)) is \(41^{\circ}\). In \(\triangle GHI\), the included angle (the angle between sides \(HI = 18\) and \(HG = 27\)) is also \(41^{\circ}\)

Answer:

Triangles \(DEF\) and \(GHI\) are similar. By the Side - Angle - Side (SAS) similarity criterion. Since the ratios of two pairs of corresponding sides are equal (\(\frac{DF}{HI}=\frac{DE}{HG}=\frac{1}{3}\)) and the included angles (\(41^{\circ}\)) are equal.