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.)
sss: three sides proportionate
sss: three sides congruent
sas: two sides proportionate, included angle congruent
sas: two sides + included angle congruent
aa: two angles congruent

Explanation:

Step1: Calculate the ratio of sides

For triangle \(IJK\) and \(LMN\), we have \(\frac{IJ}{MN}=\frac{18}{90}=\frac{1}{5}\) and \(\frac{JK}{ML}=\frac{19}{95}=\frac{1}{5}\)

Step2: Check the included angle

The included angle \(\angle J = 37^{\circ}\) and \(\angle M=37^{\circ}\), so the included angles are congruent.

Since two sides are proportionate (\(\frac{IJ}{MN}=\frac{JK}{ML}=\frac{1}{5}\)) and the included angle (\(\angle J\) and \(\angle M\)) is congruent, we can use the SAS (Side - Angle - Side) similarity criterion.

Answer:

Triangle \(IJK\) and triangle \(LMN\) are similar. We know this by the SAS (two sides proportionate, included angle congruent) similarity criterion.