Question

1. refer to the triangles.

a. find the length of ef.
units
b. find the measure of angle e.
degrees

Explanation:

Step1: Determine if triangles are similar

In \(\triangle ABC\), \(\angle C=180^{\circ}-(82^{\circ} + 55^{\circ})=43^{\circ}\). In \(\triangle DEF\), \(\angle D = 55^{\circ}\), \(\angle F=43^{\circ}\). So, \(\triangle ABC\sim\triangle DEF\) (by AA similarity, as two angles are equal: \(\angle A=\angle D = 55^{\circ}\), \(\angle C=\angle F=43^{\circ}\)).

Step2: Use similarity ratio for side \(EF\)

For similar triangles \(\triangle ABC\) and \(\triangle DEF\), the ratio of corresponding sides is \(\frac{AB}{DE}=\frac{BC}{EF}\). \(AB = 12\), \(DE = 21\), \(BC = 14\). Let \(EF=x\). Then \(\frac{12}{21}=\frac{14}{x}\). Cross - multiply: \(12x=21\times14\), \(12x = 294\), \(x=\frac{294}{12}=24.5\).

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

Since \(\triangle ABC\sim\triangle DEF\), corresponding angles are equal. In \(\triangle ABC\), \(\angle B = 82^{\circ}\). So, \(\angle E=\angle B = 82^{\circ}\) (corresponding angles of similar triangles).

Answer:

a. \(24.5\) units
b. \(82\) degrees