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. (lesson 2 - 2) 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.

Explanation:

Step1: Recall rotation properties

A 180 - degree rotation is a rigid transformation. Rigid transformations preserve side - lengths and angle - measures, which means the pre - image and the image are congruent. Here, $\triangle FGH$ and $\triangle HEF$ are related by a 180 - degree rotation around point $K$. So, $\triangle FGH\cong\triangle HEF$.

Step2: Analyze congruent parts

• For congruent triangles $\triangle FGH$ and $\triangle HEF$, corresponding sides and angles are congruent.
• For option A: $\triangle HGF$ and $\triangle FEH$ are not in the correct corresponding order for the rotation - congruence, so this is false.
• For option B: $\triangle GFH$ and $\triangle EFH$ are not congruent as they are not corresponding triangles from the rotation, so this is false.
• For option C: Since $\triangle FGH$ is rotated 180 degrees around $K$ to get $\triangle HEF$, points $H$, $K$ are on the rotation center. $\angle KHE$ and $\angle KHG$ are supplementary and equal in measure (because of the 180 - degree rotation), so $\angle KHE\cong\angle KHG$, this is true.
• For option D: $\angle GHK$ and $\angle EFK$ are not corresponding angles in the congruent triangles $\triangle FGH$ and $\triangle HEF$, so this is false.
• For option E: $EH$ and $GH$ are corresponding sides of congruent triangles $\triangle HEF$ and $\triangle FGH$ (by the rotation), so $EH\cong GH$, this is true.
• For option F: $HG$ and $FE$ are corresponding sides of congruent triangles $\triangle FGH$ and $\triangle HEF$, so $HG\cong FE$, this is true.
• For option G: $FH$ and $HF$ are the same segment, so $FH\cong HF$ (reflexive property of congruence), this is true.

Answer:

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 $HF$.