Question

a. find the length of ef.
type your answer in the box.
units
b. find the measure of angle e.
type your answer in the box.

Explanation:

Step1: Determine similarity of triangles

Triangles \(ABC\) and \(DEF\) are similar. Because \(\angle A=\angle D = 60^{\circ}\), and we can find \(\angle B=180^{\circ}-84^{\circ}-36^{\circ}=60^{\circ}\), \(\angle C = 36^{\circ}\). For \(\triangle DEF\), \(\angle D=60^{\circ}\), and the ratio of sides: \(\frac{AD}{DE}=\frac{12.7}{25.4}=\frac{1}{2}\), \(\frac{AB}{DF}=\frac{7.5}{15}=\frac{1}{2}\). By the Side - Angle - Side (SAS) similarity criterion, \(\triangle ABC\sim\triangle DFE\) (the order of similarity is important for corresponding parts).

Step2: Find the length of \(EF\)

Since \(\triangle ABC\sim\triangle DFE\), the ratio of corresponding sides is \(k = 2\). The side \(BC\) corresponds to \(EF\). Given \(BC = 11.1\), then \(EF=2\times BC\).

$$EF = 2\times11.1=22.2$$

Step3: Find the measure of \(\angle E\)

Corresponding angles of similar triangles are equal. \(\angle E\) corresponds to \(\angle B\).
\(\angle E = 84^{\circ}\)

Answer:

a. \(22.2\) units
b. \(84^{\circ}\)