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: Calculate the ratios of corresponding sides

For triangle \(CDE\) and \(FGH\), we consider the ratios of the sides.
The ratio of \(CD = 14\) to \(GH=21\) is \(\frac{CD}{GH}=\frac{14}{21}=\frac{2}{3}\).
The ratio of \(DE = 18\) to \(GF = 27\) is \(\frac{DE}{GF}=\frac{18}{27}=\frac{2}{3}\).
The ratio of \(CE = 14\) to \(FH\) (assuming correspondence based on angle - side relation, since \(\angle D=\angle G = 49^{\circ}\)):
Since \(\frac{14}{21}=\frac{18}{27}=\frac{2}{3}\)

Step2: Check the included - angle condition

We know that \(\angle D=\angle G = 49^{\circ}\).
By the Side - Angle - Side (SAS) similarity criterion, if two sides of one triangle are proportional to two sides of another triangle and the included angles are equal, then the two triangles are similar.

Answer:

Triangle \(CDE\) and triangle \(FGH\) are similar. By the Side - Angle - Side (SAS) similarity criterion, since \(\frac{CD}{GH}=\frac{DE}{GF}=\frac{2}{3}\) and \(\angle D=\angle G = 49^{\circ}\)