Question

in triangle abc, angle a is 75° and angle b is 20°. select the triangle that is similar to triangle abc. a triangle def where angle d is 75° and angle e is 20° b triangle def where angle d is 20° and angle e is 75° c triangle def where angle d is 85° and angle e is 20° d triangle def where angle d is 20° and angle f is 85°

Explanation:

Step1: Calculate angle \( C \) in \(\triangle ABC\)

The sum of angles in a triangle is \( 180^{\circ} \). So, \( \angle C=180^{\circ}-\angle A - \angle B \). Substituting \( \angle A = 75^{\circ} \) and \( \angle B = 20^{\circ} \), we get \( \angle C=180^{\circ}-75^{\circ}-20^{\circ}=85^{\circ} \).

Step2: Check similarity using AA (Angle - Angle) criterion

For two triangles to be similar, their corresponding angles must be equal.

• In option A: \(\angle D = 75^{\circ}\), \(\angle E = 20^{\circ}\). Then \(\angle F=180^{\circ}-75^{\circ}-20^{\circ}=85^{\circ}\). So, \(\triangle ABC\sim\triangle DEF\) (by AA criterion as \(\angle A=\angle D = 75^{\circ}\), \(\angle B=\angle E = 20^{\circ}\))
• In option B: \(\angle D = 20^{\circ}\), \(\angle E = 75^{\circ}\). Here the correspondence of angles (\(A\) with \(E\) and \(B\) with \(D\)) is not in the same order as per the naming of the triangles for similarity (the order of angles in the similarity statement matters for correspondence)
• In option C: \(\angle D = 85^{\circ}\), \(\angle E = 20^{\circ}\). There is no correspondence of two angles with \(\triangle ABC\)
• In option D: \(\angle D = 20^{\circ}\), \(\angle F = 85^{\circ}\). There is no correspondence of two angles with \(\triangle ABC\)

Answer:

A. triangle \(DEF\) where angle \(D\) is \(75^{\circ}\) and angle \(E\) is \(20^{\circ}\)