Question

triangle hef is the image of triangle fgh after a 180 degree rotation around point k. select all statements that must be true. (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 rotation

A 180 - degree rotation is a rigid transformation. Rigid transformations (rotations, translations, reflections) preserve the shape and size of the figure. So, if \(\triangle FGH\) is rotated 180 degrees around point \(K\) to get \(\triangle HEF\), then \(\triangle FGH\cong\triangle HEF\) (by the definition of a rigid transformation).

Step2: Congruent angles and segments

• For angles:
• When we rotate a figure 180 degrees around a point, the corresponding angles are congruent. If we consider the rotation of point \(G\) to \(E\) and \(H\) to \(F\) (or vice - versa depending on the direction of rotation), \(\angle KHE\) and \(\angle KHG\) are vertical angles (formed by the intersection of lines \(EH\) and \(GH\) at \(K\) after rotation) and vertical angles are congruent. Also, \(\angle GHK\) and \(\angle EFK\) are corresponding angles of congruent triangles \(\triangle FGH\) and \(\triangle HEF\).
• For segments:
• Since \(\triangle FGH\cong\triangle HEF\), the corresponding segments are congruent. \(HG\) corresponds to \(FE\) (by the order of the vertices in congruent triangles \(\triangle FGH\) and \(\triangle HEF\)). The statement “Segment \(FH\) is congruent to segment \(HF\)” is a tautology (a segment is always congruent to itself, but it is not a result of the rotation transformation in the context of the two - triangle relationship). The statement “Segment \(EH\) is congruent to segment \(GH\)” is incorrect because \(EH\) corresponds to \(FG\) (from the congruence \(\triangle FGH\cong\triangle HEF\)). The statement “Triangle \(HGF\) is congruent to triangle \(FEH\)” is incorrect because the order of vertices does not match the congruence from the rotation (the correct congruence is \(\triangle FGH\cong\triangle HEF\)).

Answer:

• Triangle \(GFH\) is congruent to triangle \(EFH\)
• Angle \(KHE\) is congruent to angle \(KHG\)
• Angle \(GHK\) is congruent to angle \(EFK\)
• Segment \(HG\) is congruent to segment \(FE\)