Question

triangle hef is the image of triangle fgh after a 180 degree rotation around point k. select all statements that must be true. incorrect (select all that apply.) triangle hgf is congruent to triangle feh triangle gfh is congruent to triangle efh angle khe is congruent to angle khg angle ghk is congruent to angle efk. segment eh is congruent to segment gh segment hg is congruent to segment fe. segment fh is congruent to segment hf.

Explanation:

Step1: Properties of 180 - degree rotation

A 180 - degree rotation is a rigid transformation. Rigid transformations (rotations, translations, reflections) preserve the shape and size of the figure. So, if \(\triangle HEF\) is the image of \(\triangle FGH\) after a 180 - degree rotation around point \(K\), then \(\triangle FGH\cong\triangle HEF\) (by the property of rotation as a rigid transformation).
For angles formed by the rotation: If we consider the rotation of a point \(H\) around \(K\) to get its image \(E\) (assuming the rotation rule \((x,y)\to(-x + 2k_x,-y + 2k_y)\) for a point \((x,y)\) rotated 180 degrees around \((k_x,k_y)\)), the angles \(\angle KHE\) and \(\angle KHG\) are congruent because of the rotational symmetry.
For side - length preservation: Since rotation is a rigid transformation, the length of a segment and its image are equal. So, \(HG\) and \(FE\) are corresponding segments (pre - image and image of the rotation), so \(HG\cong FE\).

Step2: Analyzing each option

• Option 1: \(\triangle HGF\) and \(\triangle FEH\)

There is no rotation or other rigid - transformation relationship (based on the given 180 - degree rotation of \(\triangle FGH\) to \(\triangle HEF\)) that would make \(\triangle HGF\cong\triangle FEH\).

• Option 2: \(\triangle GFH\) and \(\triangle EFH\)

Since \(\triangle HEF\) is the image of \(\triangle FGH\) after a 180 - degree rotation (a rigid transformation), \(\triangle GFH\cong\triangle EFH\) (by the definition of congruence of figures under rigid transformations).

• Option 3: \(\angle KHE\) and \(\angle KHG\)

When we rotate a figure 180 degrees around a point \(K\), the angles subtended by the pre - image and image of a point (in this case, the rotation of \(H\) to \(E\) around \(K\)) with respect to \(K\) are congruent. So, \(\angle KHE\cong\angle KHG\).

• Option 4: \(\angle GHK\) and \(\angle EFK\)

There is no rotational or other geometric relationship (based on the given 180 - degree rotation of \(\triangle FGH\) to \(\triangle HEF\)) that would make \(\angle GHK\cong\angle EFK\).

• Option 5: \(EH\) and \(GH\)

\(EH\) is the image of \(FH\) (not \(GH\)) under the 180 - degree rotation (if we consider the rotation mapping of the triangle vertices).

• Option 6: \(HG\) and \(FE\)

Since rotation is a rigid transformation, the pre - image segment \(HG\) and its image segment \(FE\) (because of the 180 - degree rotation of \(\triangle FGH\) to \(\triangle HEF\)) are congruent.

• Option 7: \(FH\) and \(HF\)

\(FH\) and \(HF\) are the same segment, and while \(FH\cong FH\) (trivially true), but this is not a result of the 180 - degree rotation transformation property in the context of the problem (the non - trivial congruences based on rotation are already considered for other options).

Answer:

Triangle \(GFH\) is congruent to triangle \(EFH\), Angle \(KHE\) is congruent to angle \(KHG\), Segment \(HG\) is congruent to segment \(FE\).