Question

decide whether the triangles are similar. if they are, write a similarity statement and state the reason jus similarity. if necessary, you may learn what the markings on a figure indicate. not similar or not necessarily similar similar: by the select not similar or not necessarily similar similar: by the select not similar or not necessarily similar similar: by the select

Explanation:

First pair of triangles ($\triangle ABC$ and $\triangle DEF$)

Step1: Calculate the ratios of corresponding sides

For $\triangle ABC$ with sides \(AB = 6\), \(BC=5\), \(AC = 9\) and $\triangle DEF$ with sides \(DE = 6\), \(EF = 4\), \(DF=3\)
The ratios are \(\frac{AB}{DE}=\frac{6}{6} = 1\), \(\frac{BC}{EF}=\frac{5}{4}=1.25\), \(\frac{AC}{DF}=\frac{9}{3}=3\)
Since the ratios of the corresponding sides are not equal, \(\triangle ABC\) and \(\triangle DEF\) are not similar.

Second pair of triangles ($\triangle JKL$ and $\triangle GIH$)

Step1: Find the third - angle of \(\triangle GIH\)

In \(\triangle GIH\), using the angle - sum property of a triangle (\(\angle G+\angle I+\angle H=180^{\circ}\)), given \(\angle I = 90^{\circ}\) and \(\angle H=55^{\circ}\), then \(\angle G=180^{\circ}-(90^{\circ}+55^{\circ})=35^{\circ}\)

Step2: Check for AA (Angle - Angle) similarity

In \(\triangle JKL\), \(\angle K = 90^{\circ}\) and \(\angle L=35^{\circ}\). In \(\triangle GIH\), \(\angle I = 90^{\circ}\) and \(\angle G = 35^{\circ}\)
Two angles of \(\triangle JKL\) (\(\angle K = 90^{\circ}\) and \(\angle L = 35^{\circ}\)) are equal to two angles of \(\triangle GIH\) (\(\angle I=90^{\circ}\) and \(\angle G = 35^{\circ}\))

Third pair of triangles ($\triangle XYZ$ and $\triangle MNZ$)

Step1: Check for vertical angles

\(\angle XZY\) and \(\angle MZN\) are vertical angles. So, \(\angle XZY=\angle MZN\)
But we have no information about other angles or side - ratios. So, we cannot confirm similarity.

Answer:

1. Not similar or not necessarily similar
2. Similar: \(\triangle JKL\sim\triangle GIH\) by the AA (Angle - Angle) similarity criterion
3. Not similar or not necessarily similar