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 \( \triangle ABC \)

By the angle - sum property of a triangle (\( A + B + C=180^{\circ} \)), we have \( C = 180-(35 + 20)=125^{\circ} \)

Step2: Analyze each option

• Option A:

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

• Option B:

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

• Option C:

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

• Option D:

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

• Option E:

In \( \triangle PQR \), \( Q = 20^{\circ}\), \( R = 30^{\circ}\). Then \( P=180-(20 + 30)=130^{\circ}\). 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 \( 125^{\circ}\)