Question

the triangles are similar by
the sss similarity theorem

the sas similarity theorem

the aa similarity postulate

the triangles are not similar

Explanation:

Step1: Find ratios of corresponding sides

For triangle \(JKL\) and \(XYZ\) (assuming \(Z\) is the third vertex, let's match sides:

• \(JK = x\), \(XY = 1.5x\), ratio: \(\frac{JK}{XY}=\frac{x}{1.5x}=\frac{2}{3}\)
• \(KL = 2x\), \(XZ = 3x\), ratio: \(\frac{KL}{XZ}=\frac{2x}{3x}=\frac{2}{3}\)
• \(JL=\frac{2}{3}y\), \(YZ = y\), ratio: \(\frac{JL}{YZ}=\frac{\frac{2}{3}y}{y}=\frac{2}{3}\)

Step2: Apply SSS similarity

SSS similarity theorem states that if the ratios of all three corresponding sides of two triangles are equal, the triangles are similar. Here, all three ratios are \(\frac{2}{3}\), so by SSS similarity theorem.

Answer:

the SSS similarity theorem