Question

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

a. triangle hgf is congruent to triangle feh.
b. triangle gfh is congruent to triangle efh.
c. angle khe is congruent to angle khg.
d. angle ghk is congruent to angle efk.
e. segment eh is congruent to segment gh.
f. segment hg is congruent to segment fe.
g. segment fh is congruent to segment hf.
(from unit 2, lesson 2.)

Explanation:

Step1: Properties of rotation

A \(180^{\circ}\) rotation is a rigid transformation. Rigid transformations preserve side - lengths and angle - measures. So, if \(\triangle HEF\) is the image of \(\triangle FGH\) after a \(180^{\circ}\) rotation around point \(K\), then \(\triangle FGH\cong\triangle HEF\) (by the definition of congruence in rigid transformations).

Step2: Analyze option A

For \(\triangle HGF\) and \(\triangle FEH\), there is no information from the rotation transformation that would imply they are congruent. The order of vertices and side - length/angle - measure relationships are not satisfied for congruence.

Step3: Analyze option B

Since \(\triangle FGH\) is rotated \(180^{\circ}\) to get \(\triangle HEF\), \(\triangle GFH\) (same as \(\triangle FGH\)) and \(\triangle EFH\) (same as \(\triangle HEF\)) are congruent.

Step4: Analyze option C

Because of the \(180^{\circ}\) rotation, \(H\), \(K\), \(E\) are collinear and \(H\), \(K\), \(G\) are collinear. \(\angle KHE\) and \(\angle KHG\) are vertical angles (formed by the intersection of lines \(HE\) and \(HG\) at \(K\)). Vertical angles are congruent.

Step5: Analyze option D

There is no reason from the rotation transformation to suggest \(\angle GHK\cong\angle EFK\). The angles are not in corresponding positions of the rotated triangles.

Step6: Analyze option E

Since \(\triangle FGH\cong\triangle HEF\), \(GH = EH\) (corresponding parts of congruent triangles).

Step7: Analyze option F

Since \(\triangle FGH\cong\triangle HEF\), \(HG=FE\) (corresponding parts of congruent triangles).

Step8: Analyze option G

\(FH\) and \(HF\) are the same segment. By the reflexive property of congruence, \(FH\cong HF\).

Answer:

B. Triangle \(GFH\) is congruent to triangle \(EFH\); C. Angle \(KHE\) is congruent to angle \(KHG\); E. Segment \(EH\) is congruent to segment \(GH\); F. Segment \(HG\) is congruent to segment \(FE\); G. Segment \(FH\) is congruent to segment \(HF\)