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.

Question

The diagram shows that ∠HGI≅∠JGI
and ∠GIH≅∠GIJ
. What theorem proves that △GHI≅△GJI
?

Option # 1: SSS

Option # 2: SAS

Option # 3: AAS

Option # 4: ASA

(1 point)
△GHI≅△GJI
by Option #

Answer 1

To determine which theorem proves that triangles \( \triangle GHI \cong \triangle GJI \), we can look at the provided angle congruences and the properties of the two triangles.

Given:

• \( \angle HGI \cong \angle JGI \) (these are angles opposite the sides that are bisected by the vertical line)
• \( \angle GIH \cong \angle GIJ \) (these are angles at vertex \( G \) formed by the same line segments)

Since you have two angles congruent in each triangle and the included side \( GI \) is the same for both triangles, you can use the Angle-Side-Angle (ASA) theorem.

Thus, \( \triangle GHI \cong \triangle GJI \) by:

Option # 4: ASA

So the final answer will be: \[ \triangle GHI \cong \triangle GJI \text{ by Option # 4: ASA} \]