Question

triangle ghi is rotated 90° clockwise and then reflected over the y - axis. which congruency statement is true? △ghi ≅ △ghi △ghi ≅ △ihg ∠g ≅ ∠h ≅ ∠i ∠g ≅ ∠h ≅ ∠i

Explanation:

Step1: Understand the properties of rotation and reflection

Rotation and reflection are rigid transformations. Rigid transformations preserve the shape and size of a figure, which means that the pre - image and the image are congruent.

Step2: Analyze each option

• For \(\triangle GHI\cong\triangle G''H''I''\): Since rotation (a rigid transformation) by \(90^{\circ}\) clockwise and reflection (a rigid transformation) over the \(y -\)axis are both rigid transformations. A rotation of a triangle by \(90^{\circ}\) clockwise changes the orientation of the triangle, and then reflecting it over the \(y -\)axis. But the side lengths and angle measures of \(\triangle GHI\) and \(\triangle G''H''I''\) are the same. Because rigid transformations (rotation and reflection) do not change the size and shape of the figure.
• For \(\triangle GHI\cong\triangle IHG\): Just changing the order of the vertices does not imply congruence based on the transformation described. Congruence \(\triangle ABC\cong\triangle DEF\) means that the corresponding parts (sides and angles) are equal. Here, \(\triangle GHI\) and \(\triangle IHG\) do not have a transformation (rotation and reflection as described in the problem) relationship.
• For \(\angle G\cong\angle H\cong\angle I\): There is no information from the transformation (rotation and reflection) that would make these three angles congruent. In a general triangle (and we have no indication that \(\triangle GHI\) is equilateral from the transformation description), angles are not necessarily congruent.
• For \(\angle G''\cong\angle H''\cong\angle I''\): Similar to the previous non - congruence of angles in \(\triangle GHI\), there is no basis from the rotation and reflection transformation to conclude that these three angles in \(\triangle G''H''I''\) are congruent.

Answer:

\(\triangle GHI\cong\triangle G''H''I''\)