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: Check the ratio of corresponding sides

For side - length comparison: $\frac{IJ}{MN}=\frac{8}{32}=\frac{1}{4}$, $\frac{JK}{ML}=\frac{6}{24}=\frac{1}{4}$, $\frac{KI}{LN}=\frac{6}{24}=\frac{1}{4}$

Step2: Check the included - angle

The included angle $\angle J = 49^{\circ}$ and $\angle M=49^{\circ}$, so the included angles are equal.
Since the ratios of the corresponding sides are equal ($\frac{IJ}{MN}=\frac{JK}{ML}=\frac{KI}{LN}=\frac{1}{4}$) and the included angles are equal ($\angle J=\angle M = 49^{\circ}$), by the Side - Angle - Side (SAS) similarity criterion, the two triangles are similar.

Answer:

The triangles $\triangle IJK$ and $\triangle LMN$ are similar. We know this by the Side - Angle - Side (SAS) similarity criterion because the ratios of the corresponding sides are equal ($\frac{IJ}{MN}=\frac{JK}{ML}=\frac{KI}{LN}=\frac{1}{4}$) and the included angles $\angle J$ and $\angle M$ are equal ($\angle J=\angle M = 49^{\circ}$).