Geometry worksheet featuring a two-column proof template for proving segment properties in a non-isosceles triangle.

- Given: In △GHF, ∠GIF ≅ ∠GIH and △GHF is not isosceles.
- Prove: Segment GI is not a median.

| Statement | Reason |
|-----------|--------|
| 1. ∠GIF ≅ ∠GIH | 1. Given |
| 2. GI ≅ GI | 2. Reflexive Property of Congruence |
| 3. △GIF ≅ △GIH | 3. ASA Congruence Postulate (Statements 1, 2, and the shared side GI) |
| 4. FI ≅ HI | 4. CPCTC (Corresponding Parts of Congruent Triangles are Congruent) |
| 5. I is the midpoint of FH | 5. Definition of Midpoint (from Statement 4) |
| 6. GI is a median of △GHF | 6. Definition of Median (a segment from a vertex to the midpoint of the opposite side) |
| 7. △GHF is not isosceles | 7. Given |
| 8. GF ≠ GH | 8. Definition of Not Isosceles (no two sides are congruent) |
| 9. Contradiction: From Statements 3 and 8, △GIF ≅ △GIH implies GF ≅ GH, which contradicts Statement 8. | 9. Contradiction from assuming GI is a median |
| 10. Therefore, GI is not a median. | 10. Proof by Contradiction (the assumption that GI is a median leads to a contradiction with the given that △GHF is not isosceles) |