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△EFG is rotated −90 degrees (clockwise) to create △XYZ . Which of the following statements must be true?(1 point)

Responses which is the only correct answer that is 100 percent correct and why solve it and give me explanation

EG¯¯¯¯¯¯¯¯≅XY¯¯¯¯¯¯¯¯
Modifying above upper E upper G with bar congruent to Modifying above upper X upper Y with bar

EF¯¯¯¯¯¯¯¯≅XZ¯¯¯¯¯¯¯¯
Modifying above upper E upper F with bar congruent to Modifying above upper X upper Z with bar

∠F≅∠Z
angle upper F congruent to angle upper Z

∠E≅∠X
angle upper E congruent to angle upper X

1 answer

When triangle \( \triangle EFG \) is rotated -90 degrees (clockwise), we can analyze how the vertices and sides correspond in triangle \( \triangle XYZ \).

To understand what this rotation means, we need to consider the properties of rotations in geometry:

  1. Preservation of Lengths: Rotations preserve the lengths of segments. Hence, if two segments in the original triangle are equal, the corresponding segments in the rotated triangle will also be equal.

  2. Preservation of Angles: Rotations preserve the measures of angles. Thus, corresponding angles in the rotated triangle will also be equal.

Given these properties, let’s analyze each statement:

  1. \( \overline{EG} \cong \overline{XY} \):

    • Since \( \overline{EG} \) is a side of \( \triangle EFG \) and corresponds to the segment connecting the points after the rotation, this statement could be true depending on how the triangle is oriented before and after rotation.

  2. \( \overline{EF} \cong \overline{XZ} \):

    • This statement also holds true as the sides will remain congruent after the rotation.

  3. \( \angle F \cong \angle Z \):

    • When you rotate the triangle, the angles remain congruent. Therefore, this statement must be true.

  4. \( \angle E \cong \angle X \):

    • Similar to the previous point, this statement is also true as rotating the triangle maintains the measure of angles.

From the nature of rotations, the statement that must be universally true and is most direct is \( \angle F \cong \angle Z \). This is because the angles are kept congruent no matter how the triangle is rotated.

The correct answer is \( \angle F \cong \angle Z \) because upon rotation, all angles of the triangle remain congruent. Each individual angle's measure does not change, thus making this statement unequivocally true.