Question

1. triangle hef is the image of triangle fgh after a 180 degree rotation around point k. select 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.

Explanation:

Step1: Properties of 180 - degree rotation

A 180 - degree rotation is a transformation. When a figure is rotated 180 degrees around a point, the pre - image and the image are congruent. For two congruent triangles (pre - image and image after 180 - degree rotation), corresponding angles and corresponding sides are congruent.

Step2: Analyze each option

• Option a:

When we rotate \(\triangle HGF\) 180 degrees around point \(K\) to get \(\triangle HEF\), \(\angle HGF\) and \(\angle HEF\) are not corresponding angles. So, \(\triangle HGF\) is not congruent to \(\triangle FEH\).

• Option b:

When we rotate \(\triangle GFH\) 180 degrees around point \(K\) to get \(\triangle EFH\), \(\angle GFH\) and \(\angle EFH\) are not corresponding angles. So, \(\triangle GFH\) is not congruent to \(\triangle EFH\).

• Option c:

When we rotate \(\triangle KHE\) 180 degrees around point \(K\) to get \(\triangle KHG\), \(\angle KHE\) and \(\angle KHG\) are not corresponding angles. So, \(\angle KHE\) is not congruent to \(\angle KHG\).

• Option d:

When we rotate \(\triangle GHK\) 180 degrees around point \(K\) to get \(\triangle EF K\), \(\angle GHK\) and \(\angle EF K\) are not corresponding angles. So, \(\angle GHK\) is not congruent to \(\angle EF K\).

• Option e:

When we rotate segment \(EH\) 180 degrees around point \(K\), the image of \(EH\) is \(GH\). Since rotation is a rigid transformation (preserves length), segment \(EH\) is congruent to segment \(GH\).

• Option f:

When we rotate segment \(HG\) 180 degrees around point \(K\), the image of \(HG\) is \(EF\), not \(FE\).

• Option g:

When we rotate segment \(FH\) 180 degrees around point \(K\), the image of \(FH\) is itself (because the center of rotation \(K\) lies on \(FH\) and rotation of a point on the line of rotation by 180 degrees about a point on the line gives the same line segment). So, segment \(FH\) is congruent to segment \(HF\).

Answer:

e. Segment \(EH\) is congruent to segment \(GH\); g. Segment \(FH\) is congruent to segment \(HF\)