Question

triangle ghi and triangle qrs are drawn below. if \\( \overline { g i } \cong \overline { q s } , \angle i \cong \angle s \\), and \\( \angle g \cong \angle q \\), use the dropdown boxes below to determine a sequence of two transformations that maps triangle ghi onto triangle qrs. then use a congruence statement to explain why this is possible.

reflect triangle ghi over \\( \overline { h i } \\). then translate triangle \\( g ^ { prime } h ^ { prime } i ^ { prime } \\) until point \\( i ^ { prime } \\) aligns with point s.

triangle ghi maps onto triangle qrs. this is possible because \\( \triangle g h i \cong \triangle q r s \\) by aas, and one congruent figure can be mapped onto another using rigid motions.

Explanation:

Step1: Analyze the given congruent parts

We are given that \(\overline{GI}\cong\overline{QS}\), \(\angle I\cong\angle S\), and \(\angle G\cong\angle Q\). This satisfies the Angle - Angle - Side (AAS) congruence criterion. In the AAS congruence, if two angles and a non - included side of one triangle are congruent to two angles and the corresponding non - included side of another triangle, then the triangles are congruent.

Step2: Consider the transformation

First, reflecting triangle \(GHI\) over \(\overline{HI}\) changes its orientation. Then, translating (sliding) the reflected triangle \(G'H'I'\) (after reflection) until point \(I'\) (the image of \(I\) after reflection) aligns with point \(S\). Since \(\triangle GHI\cong\triangle QRS\) (by AAS), the two - step transformation (reflection followed by translation) will map \(\triangle GHI\) onto \(\triangle QRS\) because rigid motions (reflection and translation) preserve the shape and size of a figure. Rigid motions are transformations that do not change the lengths of sides or measures of angles of a figure.

Answer:

The sequence of transformations is: Reflect triangle \(GHI\) over \(\overline{HI}\), then translate the reflected triangle. The congruence statement is \(\triangle GHI\cong\triangle QRS\) by AAS (Angle - Angle - Side) because \(\overline{GI}\cong\overline{QS}\), \(\angle I\cong\angle S\), and \(\angle G\cong\angle Q\), and rigid motions (reflection and translation) can map one congruent triangle onto another.