Question

in triangle abc, angle a is 35° and angle b is 20°. select all triangles which are similar to triangle abc. a triangle def where angle d is 35° and angle e is 20° b triangle ghi where angle g is 35° and angle i is 30° c triangle jkl where angle j is 35° and angle l is 125° d triangle mno where angle n is 20° and angle o is 125° e triangle pqr where angle q is 20° and angle r is 30°

Explanation:

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

The sum of angles in a triangle is \( 180^{\circ} \). So, \( \angle C=180^{\circ}-\angle A - \angle B=180^{\circ}-35^{\circ}-20^{\circ}=125^{\circ} \)

Step2: Check similarity for each option

• Option A:

In \( \triangle DEF \), \( \angle D = 35^{\circ},\angle E=20^{\circ} \). Then \( \angle F=180^{\circ}-35^{\circ}-20^{\circ}=125^{\circ} \). By AA (Angle - Angle) similarity criterion (\( \angle A=\angle D = 35^{\circ},\angle B=\angle E = 20^{\circ} \)), \( \triangle ABC\sim\triangle DEF \)

• Option B:

In \( \triangle GEI \), \( \angle G = 35^{\circ},\angle I=30^{\circ} \). Then \( \angle E=180^{\circ}-35^{\circ}-30^{\circ}=115^{\circ} \). Since the angles are not equal to the angles of \( \triangle ABC \), \( \triangle ABC \) and \( \triangle GEI \) are not similar.

• Option C:

In \( \triangle JKL \), \( \angle J = 35^{\circ},\angle L=125^{\circ} \). Then \( \angle K=180^{\circ}-35^{\circ}-125^{\circ}=20^{\circ} \). By AA similarity criterion (\( \angle A=\angle J = 35^{\circ},\angle C=\angle L = 125^{\circ} \)), \( \triangle ABC\sim\triangle JKL \)

• Option D:

In \( \triangle MNO \), \( \angle N = 20^{\circ},\angle O=125^{\circ} \). Then \( \angle M=180^{\circ}-20^{\circ}-125^{\circ}=35^{\circ} \). By AA similarity criterion (\( \angle B=\angle N = 20^{\circ},\angle C=\angle O = 125^{\circ} \)), \( \triangle ABC\sim\triangle MNO \)

• Option E:

In \( \triangle PQR \), \( \angle Q = 20^{\circ},\angle R=30^{\circ} \). Then \( \angle P=180^{\circ}-20^{\circ}-30^{\circ}=130^{\circ} \). Since the angles are not equal to the angles of \( \triangle ABC \), \( \triangle ABC \) and \( \triangle PQR \) are not similar.

Answer:

A. triangle \( DEF \) where angle \( D \) is \( 35^{\circ} \) and angle \( E \) is \( 20^{\circ} \)

C. triangle \( JKL \) where angle \( J \) is \( 35^{\circ} \) and angle \( L \) is \( 125^{\circ} \)

D. triangle \( MNO \) where angle \( N \) is \( 20^{\circ} \) and angle \( O \) is \( 125^{\circ} \)