To determine which of the statements must be true after triangle \( \triangle EFG \) is rotated \(-90^\circ\) (clockwise) to create triangle \( \triangle XYZ \), we can analyze the relationships of the vertices and the sides based on the properties of rotation.
When triangle \( \triangle EFG \) is rotated \(-90^\circ\) (clockwise), the corresponding vertices transform as follows:
- Point \( E \) moves to point \( Z \).
- Point \( F \) moves to point \( X \).
- Point \( G \) moves to point \( Y \).
From this transformation, we can determine the following relationships:
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Congruent Sides: Since rotation preserves distances,
- \( \overline{EF} \cong \overline{XZ} \)
- \( \overline{FG} \cong \overline{XY} \)
- \( \overline{EG} \cong \overline{YZ} \)
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Congruent Angles: Since the angles are preserved under rotation,
- \( \angle EFG \cong \angle ZXY \)
- \( \angle FGE \cong \angle XZY \)
- \( \angle GEF \cong \angle YXZ \)
With these transformations, we can evaluate the provided statements:
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Statement 1: \(\overline{EF} \cong \overline{XZ}\)
- This statement is true.
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Statement 2: \(\angle F \cong \angle Z\)
- This statement is false; according to the vertex mappings, \( \angle F \) corresponds to \( \angle X \).
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Statement 3: \(\angle E \cong \angle X\)
- This statement is false; \( \angle E \) corresponds to \( \angle Z\).
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Statement 4: \(\overline{EG} \cong \overline{XY}\)
- This statement is also false.
Based on this analysis, the only statement that must be true is the first one. Thus, the final answer is:
Only Statement 1 is true.