Question

determine if triangle ijk and triangle lmn are or are not similar, and, if they are, state how you know. (note that figures are not necessarily drawn to scale.)

Explanation:

Step1: Calculate the ratios of corresponding sides

For triangle \(IJK\) and \(LMN\), we have:

• \(\frac{JK}{MN}=\frac{16}{32}=\frac{1}{2}\)
• \(\frac{KI}{NL}=\frac{18}{36}=\frac{1}{2}\)

Step2: Check the included angle

The included angle \(\angle K = \angle N=39^{\circ}\)

Step3: Apply the SAS (Side - Angle - Side) similarity criterion

Since the ratios of two pairs of corresponding sides are equal (\(\frac{JK}{MN}=\frac{KI}{NL}=\frac{1}{2}\)) and the included angles (\(\angle K\) and \(\angle N\)) are equal, by the SAS similarity criterion, the two triangles are similar.

Answer:

Triangle \(IJK\) and triangle \(LMN\) are similar. We know this by the SAS (Side - Angle - Side) similarity criterion. The ratio of \(JK\) to \(MN\) is \(\frac{1}{2}\), the ratio of \(KI\) to \(NL\) is \(\frac{1}{2}\), and \(\angle K=\angle N = 39^{\circ}\)