Question

in triangles ( jkl ) and ( mno ), angles ( j ) and ( m ) each have measure ( 30^{circ} ), side ( jk = 4 ), and side ( mn = 12 ). which additional piece of information is sufficient to prove that triangles ( jkl ) and ( mno ) are similar?
a side ( jl = 8 ) and side ( mo = 8 )
b side ( kl = 8 ) and side ( mo = 24 )
c angle ( k = 30^{circ} ) and angle ( o = 60^{circ} )
d angle ( k = 90^{circ} ) and angle ( n = 90^{circ} )

Explanation:

Step1: Recall the AA (Angle - Angle) similarity criterion

Two triangles are similar if two pairs of corresponding angles are equal.

Step2: Analyze option D

We know that \(\angle J=\angle M = 30^{\circ}\). If \(\angle K = 90^{\circ}\) and \(\angle N=90^{\circ}\), then in \(\triangle JKL\), \(\angle L=180^{\circ}-\angle J-\angle K=180^{\circ}- 30^{\circ}-90^{\circ}=60^{\circ}\). In \(\triangle MNO\), \(\angle O=180^{\circ}-\angle M-\angle N=180^{\circ}-30^{\circ}-90^{\circ}=60^{\circ}\). So, \(\angle J=\angle M\), \(\angle K=\angle N\), and \(\angle L=\angle O\). By AA similarity criterion, \(\triangle JKL\sim\triangle MNO\).

Step3: Analyze option A

If \(JL = 8\) and \(MO = 8\), we have \(JK = 4\), \(MN=12\), \(JL = 8\), \(MO = 8\). \(\frac{JK}{MN}=\frac{4}{12}=\frac{1}{3}\), \(\frac{JL}{MO}=\frac{8}{8} = 1\). The ratios of sides are not equal and we don't have information about angles, so we can't prove similarity.

Step4: Analyze option B

If \(KL = 8\) and \(MO = 24\), we know \(JK = 4\), \(MN = 12\). \(\frac{JK}{MN}=\frac{4}{12}=\frac{1}{3}\), but we don't know the included angles between the sides. So, we can't prove similarity using SAS (Side - Angle - Side) similarity (since we don't know if the included angles are equal).

Step5: Analyze option C

If \(\angle K=30^{\circ}\) and \(\angle O = 60^{\circ}\), in \(\triangle JKL\), \(\angle L=180^{\circ}-\angle J-\angle K=180^{\circ}-30^{\circ}-30^{\circ}=120^{\circ}\). In \(\triangle MNO\), \(\angle M = 30^{\circ}\), \(\angle O=60^{\circ}\), then \(\angle N=180^{\circ}-\angle M-\angle O=90^{\circ}\). The angles are not equal in pairs, so we can't prove similarity.

Answer:

D. Angle \(K = 90^{\circ}\) and angle \(N = 90^{\circ}\)