Question

conditions for triangle similarity

1. what is the length of segment df?

68°
44°
8
6
68°
44°

Explanation:

Step1: Determine similarity of triangles

In \(\triangle ABC\) and \(\triangle DEF\), \(\angle A = 68^{\circ}\), \(\angle B=44^{\circ}\), \(\angle D = 68^{\circ}\), \(\angle F = 44^{\circ}\). By the AA (Angle - Angle) similarity criterion, \(\triangle ABC\sim\triangle DEF\).

Step2: Set up proportion

Since \(\triangle ABC\sim\triangle DEF\), the ratios of corresponding sides are equal. \(\frac{AB}{DE}=\frac{AC}{DF}\). We know \(AB = 8\), \(DE=18\), \(AC = 6\). Substituting into the proportion: \(\frac{8}{18}=\frac{6}{DF}\).

Step3: Solve for \(DF\)

Cross - multiply: \(8\times DF=18\times6\). So \(8DF = 108\). Then \(DF=\frac{108}{8}=\frac{27}{2}=13.5\).

Answer:

\(13.5\)