Question

it triangle egf is the image of triangle fgh after a 180 degree rotation about point k. select all statements that are 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.

1. segment gh is congruent to triangle ade? explain your reasoning.

Explanation:

Step1: Properties of 180 - degree rotation

A 180 - degree rotation about a point is a transformation. If \(\triangle FEH\) is rotated 180 degrees about point \(K\) to get \(\triangle FGH\), then corresponding parts of the two triangles are congruent. Also, for a 180 - degree rotation, if we consider the rotation of a segment, the length of the segment remains the same.

Step2: Analyzing each option

• Option (a):

When we rotate \(\triangle FEH\) 180 degrees about \(K\) to get \(\triangle FGH\), by the property of rotation (a rigid transformation), \(\triangle FEH\cong\triangle GFH\) is incorrect. The correct correspondence should be based on the rotation rule.

• Option (b):

Since \(\triangle FEH\) is rotated 180 degrees about \(K\) to get \(\triangle FGH\), \(\angle KHE\) and \(\angle KFG\) are not corresponding angles. For a 180 - degree rotation, if we consider the rotation of \(\triangle FEH\) around \(K\), the correct angle - congruence relation is based on the rotation of the whole triangle.

• Option (c):

For a 180 - degree rotation about \(K\), \(\angle GHK\) and \(\angle KHE\) are not congruent. The rotation of \(\triangle FEH\) about \(K\) by 180 degrees swaps the positions of the vertices in a specific way.

• Option (d):

When we rotate a segment, for example, segment \(EH\) in \(\triangle FEH\) is rotated 180 degrees about \(K\) to get segment \(FG\). Since rotation is a rigid transformation (preserves length), \(EH\cong FG\).

• Option (e):

Segment \(GH\) is not congruent to segment \(EF\). When we perform a 180 - degree rotation of \(\triangle FEH\) about \(K\), the correspondence of segments is based on the rotation of the vertices.

Answer:

d. Segment \(EH\) is congruent to segment \(FG\).