Your proof demonstrates that line segment JK divides the sides of triangle GHI proportionally by establishing the similarity of triangles GJI and HKI. Here's a breakdown of how your argument works:
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Parallel Lines: You start by asserting that line JK is parallel to line GH. This is a critical observation as it sets the foundation for the application of the properties of parallel lines and transversals.
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Angle Relationships: Because JK is parallel to GH, you correctly conclude that the corresponding angles formed by the transversal JI are equal. Specifically, you note that \( \angle GJI = \angle HKI \). This relationship is a direct consequence of the Alternate Interior Angles Theorem, which states that when a transversal crosses parallel lines, each pair of alternate interior angles is equal.
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Triangle Similarity: You apply the Angle-Angle Similarity (AAS) Criterion. Since you have established that one pair of angles is equal (\(\angle GJI = \angle HKI\)), the similarity criterion can be further substantiated by recognizing that the angle at J (common to both triangles) is also equal. By confirming that these angles are equal, you can infer that triangles GJI and HKI are similar.
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Proportionality of Sides: When two triangles are similar, their corresponding sides are proportional. This means that the ratios of the lengths of their sides are equal. Hence, from the similarity of triangles GJI and HKI, you can conclude that:
\[ \frac{GJ}{JI} = \frac{HK}{KI} \]
This proportionality is the crux of your proof—indicating that line JK divides the sides GJ and JI of triangle GHI proportionally in relation to HK and KI.
Your proof is well-structured and clearly shows the logical progression from the properties of parallel lines to the conclusion about the proportionality of the sides. Good job!