Question

1. triangle hef is the image of triangle fgh after a 180° rotation around 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 eh is congruent to segment fg. f. segment gh is congruent to segment ef.

Explanation:

Step1: Properties of rotation

A \(180^{\circ}\) rotation is a rigid transformation. Rigid transformations preserve side - lengths and angle - measures. So, \(\triangle HEF\cong\triangle FGH\) (by the definition of a rotation as a rigid transformation).

Step2: Analyze each option

• Option A:

Since \(\triangle HEF\) is the image of \(\triangle FGH\) after a \(180^{\circ}\) rotation (a rigid transformation), \(\triangle FGH\cong\triangle HEF\) (not \(\triangle FEH\)). So, this option is false.

• Option B:

Because \(\triangle HEF\) is the image of \(\triangle FGH\) after a \(180^{\circ}\) rotation (a rigid transformation), \(\triangle EFH\cong\triangle GFH\) (corresponding parts of congruent triangles).

• Option C:

\(\angle KHE\) and \(\angle KFG\) are not corresponding angles in the congruent triangles formed by the rotation. So, this option is false.

• Option D:

Since \(H\) is rotated \(180^{\circ}\) around \(K\), \(\angle GHK\) and \(\angle KHE\) are vertical angles (formed by the intersection of lines \(HG\) and \(HE\) at \(H\) after rotation). Vertical angles are congruent.

• Option E:

Since \(\triangle HEF\cong\triangle FGH\), by the property of congruent triangles (corresponding parts of congruent triangles are congruent), \(EH = FG\).

• Option F:

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

Answer:

B. Triangle \(EFH\) is congruent to triangle \(GFH\); D. Angle \(GHK\) is congruent to angle \(KHE\); E. Segment \(EH\) is congruent to segment \(FG\); F. Segment \(GH\) is congruent to segment \(EF\)