Question

which triangles are similar to △abc?
choose 1 answer:
△jkl only
△mnp only
both
neither

Explanation:

Step1: Calculate the angles of $\triangle ABC$

In $\triangle ABC$, we know that the sum of angles in a triangle is $180^{\circ}$. Given two angles $\angle B = 30^{\circ}$ and $\angle A=60^{\circ}$, then $\angle C=180^{\circ}-\angle A - \angle B=180^{\circ}-60^{\circ}-30^{\circ} = 90^{\circ}$.

Step2: Analyze $\triangle JKL$

In $\triangle JKL$, $\angle J = 30^{\circ}$ and $\angle L=60^{\circ}$, then $\angle K=180^{\circ}-\angle J-\angle L=180^{\circ}-30^{\circ}-60^{\circ}=90^{\circ}$. 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), since $\angle A=\angle L = 60^{\circ}$ and $\angle B=\angle J=30^{\circ}$, $\triangle ABC\sim\triangle JKL$.

Step3: Analyze $\triangle MNP$

In $\triangle MNP$, $\angle M = 40^{\circ}$ and $\angle P = 60^{\circ}$, then $\angle N=180^{\circ}-\angle M-\angle P=180^{\circ}-40^{\circ}-60^{\circ}=80^{\circ}$. Since there is no pair of equal angles between $\triangle ABC$ and $\triangle MNP$ (angles of $\triangle ABC$ are $30^{\circ},60^{\circ},90^{\circ}$ and angles of $\triangle MNP$ are $40^{\circ},60^{\circ},80^{\circ}$), $\triangle ABC$ and $\triangle MNP$ are not similar.

Answer:

$\triangle JKL$ only