Question

these figures are congruent. what is ge?
□ miles

Explanation:

Step1: Recall Congruent Triangles

Congruent triangles have corresponding sides equal. So we need to find the corresponding side to \( GE \) in \( \triangle ABC \).

Step2: Analyze Angles in \( \triangle ABC \)

In \( \triangle ABC \), angle at \( A = 80^\circ \), side \( AB = 50 \) mi, \( AC = 40 \) mi, \( BC = 58 \) mi. Sum of angles in a triangle is \( 180^\circ \), so angle at \( B = 180 - 80 - \text{angle at } C \). But in \( \triangle GEF \), angles are \( 58^\circ \) (at \( G \)) and \( 42^\circ \) (at \( F \)), so angle at \( E = 180 - 58 - 42 = 80^\circ \). So \( \triangle ABC \cong \triangle GFE \) (by angle - angle - side, since \( \angle A=\angle E = 80^\circ \), \( AB = FE = 50 \) mi, and we can match angles). Then corresponding side to \( GE \) is \( BC \). Since \( BC = 58 \) mi, \( GE = 58 \) mi.

Answer:

58