Question

△abc and △def are shown below.
which statement is true?
△abc is similar to △def.
△abc is not similar to △def.
there is not enough information to determine whether the triangles are similar.

Explanation:

Step1: Calculate angle B in △ABC

The sum of angles in a triangle is \(180^\circ\). In △ABC, we know \(\angle A = 55^\circ\) and \(\angle C = 78^\circ\). So, \(\angle B = 180^\circ - 55^\circ - 78^\circ = 47^\circ\).

Step2: Compare angles of △ABC and △DEF

In △DEF, \(\angle E = 47^\circ\), \(\angle F = 78^\circ\), so \(\angle D = 180^\circ - 47^\circ - 78^\circ = 55^\circ\). Now, we have: \(\angle A = \angle D = 55^\circ\), \(\angle B = \angle E = 47^\circ\), \(\angle C = \angle F = 78^\circ\). 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), the triangles have all corresponding angles equal.

Answer:

\(\triangle ABC\) is similar to \(\triangle DEF\).