Question

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

a. triangle fgh is congruent to triangle feh.
b. triangle efh is congruent to triangle gfh.
c. angle khe is congruent to angle kfg.
d. angle ghk is congruent to angle khe.
e. segment kh is congruent to segment fg.
f. segment gh is congruent to segment ef.

Explanation:

Step1: Recall the properties of rotation

A 180 - degree rotation about a point changes the orientation of the figure. If \(\triangle DEF\) is the image of \(\triangle FGH\) after a \(180^{\circ}\) rotation about point \(K\), then corresponding parts of the pre - image (\(\triangle FGH\)) and the image (\(\triangle DEF\)) are congruent.
For triangles, if \(\triangle FGH\) is rotated \(180^{\circ}\) about \(K\) to get \(\triangle DEF\), then \(\triangle FGH\cong\triangle DEF\) (by the property of rotation: rotation is a rigid transformation, and rigid transformations preserve congruence). Also, for angles, \(\angle GHK\) and \(\angle KHE\) are vertical angles (formed by the intersection of two lines). When we rotate \(\triangle FGH\) \(180^{\circ}\) about \(K\), \(\angle GHK\) and \(\angle KHE\) are congruent.
For line segments, \(KH = KH\) (common side). When we rotate \(\triangle FGH\) \(180^{\circ}\) about \(K\), \(FG\) and \(EH\) are corresponding sides. But \(EH
eq EF\) and \(GH
eq EF\). \(\angle KHE\) and \(\angle KFG\) are not corresponding angles in the rotation transformation. \(\triangle EFH\) and \(\triangle GFH\) are not congruent as per the rotation transformation of \(\triangle FGH\) about \(K\) to get \(\triangle DEF\).

Step2: Analyze each option

• Option A: \(\triangle FGH\cong\triangle FEH\) is false. Because when we rotate \(\triangle FGH\) \(180^{\circ}\) about \(K\) to get \(\triangle DEF\), not \(\triangle FEH\).
• Option B: \(\triangle EFH\cong\triangle GFH\) is false. There is no rotation or other rigid transformation (like translation, reflection) that would make \(\triangle EFH\) and \(\triangle GFH\) congruent based on the \(180^{\circ}\) rotation about \(K\) of \(\triangle FGH\) to get \(\triangle DEF\).
• Option C: \(\angle KHE\cong\angle KFG\) is false. They are not corresponding angles in the \(180^{\circ}\) rotation of \(\triangle FGH\) about \(K\).
• Option D: \(\angle GHK\cong\angle KHE\). Since a \(180^{\circ}\) rotation about \(K\) maps \(\angle GHK\) to \(\angle KHE\) (vertical angles formed by the intersection of lines related to the rotation).
• Option E: \(EH\cong FG\). Because in a \(180^{\circ}\) rotation (a rigid transformation), corresponding sides are congruent. When we rotate \(\triangle FGH\) \(180^{\circ}\) about \(K\) to get \(\triangle DEF\), \(FG\) and \(EH\) are corresponding sides.
• Option F: \(GH\cong EF\) is false. \(GH\) and \(EF\) are not corresponding sides in the \(180^{\circ}\) rotation of \(\triangle FGH\) about \(K\).

Answer:

D. Angle \(GHK\) is congruent to angle \(KHE\), E. Segment \(EH\) is congruent to segment \(FG\)