Question

determine if triangle jkl and triangle mno are or are not similar, and, if they are, state how you know. (note that figures are not necessarily drawn to scale.)
triangle jkl: jk = 14, kl = 16, angle at k is 60°.
triangle mno: mn = 56, no = 64, om = 60, angle at n is 60°.
answer attempt 1 out of 3
the triangles similar.

Explanation:

Step1: Identify Angles and Sides

Triangle \(JKL\) has \(\angle K = 60^\circ\), \(JK = 14\), \(KL = 16\). Triangle \(MNO\) has \(\angle N = 60^\circ\), \(MN = 56\), \(NO = 64\), \(OM = 60\)? Wait, no, let's check the sides around the \(60^\circ\) angle. In \(JKL\), sides adjacent to \(60^\circ\) (at \(K\)) are \(JK = 14\) and \(KL = 16\). In \(MNO\), sides adjacent to \(60^\circ\) (at \(N\)) are \(MN = 56\) and \(NO = 64\)? Wait, no, \(MN = 56\), \(NO = 64\)? Wait, let's check ratios.

Wait, \(JK = 14\), \(KL = 16\); \(MN = 56\), \(NO = 64\)? Wait, \(56 \div 14 = 4\), \(64 \div 16 = 4\). So the ratio of the sides around the \(60^\circ\) angle is \(4\) (since \(14 \times 4 = 56\), \(16 \times 4 = 64\)). Also, the included angle is \(60^\circ\) in both. So by SAS (Side-Angle-Side) similarity criterion, if two sides are in proportion and the included angle is equal, triangles are similar.

Step2: Apply SAS Similarity

For \(\triangle JKL\) and \(\triangle MNO\):

• \(\angle K = \angle N = 60^\circ\) (included angle)
• \(\frac{JK}{MN} = \frac{14}{56} = \frac{1}{4}\)
• \(\frac{KL}{NO} = \frac{16}{64} = \frac{1}{4}\)

Since the two sides around the equal angle are in proportion (\(\frac{1}{4}\)) and the included angle is equal, by SAS similarity, the triangles are similar.

Answer:

The triangles are similar by the SAS (Side - Angle - Side) similarity criterion because the ratio of the sides around the \(60^\circ\) angle in both triangles is equal (\(\frac{14}{56}=\frac{16}{64}=\frac{1}{4}\)) and the included angle (\(60^\circ\)) is equal.