Question

given \\( \triangle b a c \cong \triangle f e d \\), find the missing side lengths and angle measures.
\\( e d = \\)
\\( f e = \\)
\\( m \angle b = \\)

Explanation:

Step1: Identify corresponding sides and angles

Since \(\triangle BAC\cong\triangle FED\), corresponding parts are equal.

• \(AC = ED\), \(BA=FE\), \(\angle B=\angle F\)
• In \(\triangle BAC\), \(AC = 21\) mi. So \(ED=21\) mi.
• In right - triangle \(BAC\), using Pythagorean theorem \(BC^{2}=AB^{2}+AC^{2}\). Also, since \(\triangle BAC\cong\triangle FED\), \(BC = FD = 29\) mi. Let \(AB=x\), then \(x^{2}+21^{2}=29^{2}\), \(x^{2}=29^{2}-21^{2}=(29 + 21)(29 - 21)=50\times8 = 400\), \(x = 20\) mi. So \(FE = 20\) mi.
• In \(\triangle BAC\), \(\angle A=90^{\circ}\), \(\angle C = 43.6^{\circ}\). Using the angle - sum property of a triangle (\(\angle A+\angle B+\angle C=180^{\circ}\)), \(\angle B=180^{\circ}-\angle A-\angle C\). Substitute \(\angle A = 90^{\circ}\) and \(\angle C=43.6^{\circ}\), \(\angle B=180^{\circ}-90^{\circ}-43.6^{\circ}=46.4^{\circ}\)

Answer:

\(ED = 21\) mi, \(FE=20\) mi, \(m\angle B = 46.4^{\circ}\)