Question

determine if triangle cde and triangle fgh 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 triangle \(CDE\) and \(FGH\), we have \(\frac{CD}{FG}=\frac{9}{36}=\frac{1}{4}\), \(\frac{CE}{GH}=\frac{8}{32}=\frac{1}{4}\), \(\frac{DE}{FH}=\frac{11}{44}=\frac{1}{4}\)

Step2: Check the included angle

The included angle \(\angle C\) in \(\triangle CDE\) is \(44^{\circ}\). Since the sides are in proportion (\(\frac{CD}{FG}=\frac{CE}{GH}=\frac{DE}{FH}=\frac{1}{4}\)) and we assume the angles between the corresponding proportional sides are congruent (by the property of similar triangles when SSS - like side - proportion is considered in the context of SAS - similarity, here we can also use the fact that if three sides of one triangle are proportional to three sides of another triangle, the triangles are similar. But if we consider the angle - side - angle concept in a more detailed way for the given problem, we note that the side - side - side proportion implies similarity).

Answer:

The triangles \(CDE\) and \(FGH\) are similar. We know this because \(\frac{CD}{FG}=\frac{CE}{GH}=\frac{DE}{FH}=\frac{1}{4}\), so by the SSS (Side - Side - Side) similarity criterion, \(\triangle CDE\sim\triangle FGH\)