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: Check side ratios

First, we find the ratios of corresponding sides. Let's pair the sides:

• \( \frac{DE}{GI} = \frac{20}{66} = \frac{10}{33} \)
• \( \frac{DF}{HI} = \frac{22}{69} \) (Wait, no, let's re - pair. Let's see the sides: DE = 20, EF = 23, FD = 22; GI = 66, IH = 69, HG = 60. So correct pairing: \( \frac{DE}{HG}=\frac{20}{60}=\frac{1}{3} \), \( \frac{EF}{IH}=\frac{23}{69}=\frac{1}{3} \), \( \frac{FD}{GI}=\frac{22}{66}=\frac{1}{3} \)

Step2: Check angle (if needed)

Since the ratios of all three corresponding sides are equal (\( \frac{1}{3} \)), by the Side - Side - Side (SSS) similarity criterion, the triangles are similar. Also, we can check the included angle. Let's assume the angle between the sides with ratios. For triangle DEF, the sides around a potential angle and for triangle GHI. But since SSS similarity is satisfied (all sides proportional), we can conclude similarity.

Answer:

Triangle \( DEF \) and triangle \( GHI \) are similar. We know this by the SSS (Side - Side - Side) similarity criterion because the ratios of their corresponding sides are equal: \( \frac{DE}{HG}=\frac{20}{60}=\frac{1}{3} \), \( \frac{EF}{IH}=\frac{23}{69}=\frac{1}{3} \), and \( \frac{FD}{GI}=\frac{22}{66}=\frac{1}{3} \).