Use the image to answer the question.

Question

A vertically oriented kite shape upper G upper H upper I upper J is bisected by a vertical line from upper G to upper I, dividing the kite into two equal triangles upper G upper H upper I and upper G upper J upper I. Angles are labeled.

The diagram shows that ∠HGI≅∠JGI
and ∠GIH≅∠GIJ
. What additional congruence statement can be made based on the diagram?

Option #1: ∠H≅∠J

Option #2: GI¯¯¯¯¯¯¯≅GI¯¯¯¯¯¯¯

Option #3: HI¯¯¯¯¯¯¯≅JI¯¯¯¯¯¯

Option #4: GH¯¯¯¯¯¯¯¯≅GJ¯¯¯¯¯¯¯

(1 point)
Option #

Answer 1

To determine the additional congruence statement that can be made based on the given information and the properties of the kite, let's analyze the options.

Given that:

• \(\angle HGI \cong \angle JGI\) (angles on the same side of the vertical line)
• \(\angle GIH \cong \angle GIJ\) (angles on the other side of the vertical line)

Since the kite is symmetrical about the vertical line through G and I, the corresponding sides adjacent to these angles must also be congruent.

Evaluating the Options:

1. Option #1: \(\angle H \cong \angle J\)

   • This is not necessarily true given the information provided.

2. Option #2: \(GI \cong GI\)

   • This is trivially true because any segment is congruent to itself, but it does not offer new information.

3. Option #3: \(HI \cong JI\)

   • This could be true because in a kite, the diagonal that bisects the kite (GI) also bisects the opposite angles (H and J are at the ends of this diagonal). However, there's not enough congruence information given about HI and JI.

4. Option #4: \(GH \cong GJ\)

   • This is correct because in a kite, the two pairs of adjacent sides are equal. Therefore, the sides adjacent to the angles formed by the diagonal are congruent.

Based on the analysis, the most suitable option that can be confirmed as additional congruence based on the properties of kites is:

Option #4: \(GH \cong GJ\).

Answer 2

it was option 2

Answer 3

Thank you for your clarification. Let's revisit the reasoning behind Option #2, which states \(GI \cong GI\).

While this statement is indeed true as it reflects the property that any segment is congruent to itself, the question likely hinges on the context of the kite shape where the sides and diagonals have specific properties. If the kite is symmetrical, the segments that meet at vertex \(G\) and vertex \(I\) being equal is a consequence of the kite's properties and the symmetry.

Given that the other options were not valid in the context of the angles and sides being described, if it is confirmed that the requirement in your context was focused on \(GI\) being congruent to itself as an additional statement after establishing the angle congruences, then Option #2 would indeed fit.

Thank you for your patience, and I appreciate your input! If you have any further questions or need clarification, feel free to ask.