Question

1. in triangle abc, ∠a = 30°, ∠b = 60°, and ∠c = 90°. in triangle def, ∠d = 30°, ∠e = 60°, and ∠f = 90°. which of the following is true? a. triangles abc and def have the same area b. triangles abc and def have the same perimeter c. triangles abc and def are congruent d. triangles abc and def are similar

Explanation:

Step1: Recall congruence and similarity criteria

Two triangles are congruent if their corresponding sides and angles are equal. Two triangles are similar if their corresponding angles are equal and the ratios of their corresponding sides are equal. Having the same area or perimeter does not necessarily mean congruence or similarity.

Step2: Analyze angle - angle - angle (AAA) similarity

Since \(\angle A=\angle D = 30^{\circ}\), \(\angle B=\angle E=60^{\circ}\) and \(\angle C=\angle F = 90^{\circ}\), by the AA (angle - angle) similarity criterion (if two angles of one triangle are equal to two angles of another triangle, the triangles are similar), triangles \(ABC\) and \(DEF\) are similar.

Answer:

d. Triangles \(ABC\) and \(DEF\) are similar