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△EFG is rotated −90 degrees (clockwise) to create △XYZ . Which of the following statements must be true?(1 point) Responses ∠F≅∠Z angle upper F congruent to angle upper Z EG¯¯¯¯¯¯¯¯≅XY¯¯¯¯¯¯¯¯ Modifying above upper E upper G with bar congruent to Modifying above upper X upper Y with bar ∠E≅∠X angle upper E congruent to angle upper X EF¯¯¯¯¯¯¯¯≅XZ¯¯¯¯¯¯¯¯

3 answers

To analyze the rotation of triangle \( \triangle EFG \) by -90 degrees (clockwise) to create triangle \( \triangle XYZ \), we need to understand how the points transform under this rotation.

  1. Understanding the rotation: A rotation of -90 degrees clockwise will change the position of the points as follows, assuming the origin (0,0) is the center of rotation:
    • Point E will move to point X.
    • Point F will move to point Y.
    • Point G will move to point Z.

Using this transformation:

  • The points with respect to their angles will behave as follows:
    • \( \angle EFG \) will become \( \angle XYZ \).
    • \( \angle EGF \) will become \( \angle YXZ \).
    • \( \angle FEG \) will become \( \angle ZXY \).
  1. Comparing angles and sides: After a -90 degrees rotation, the following can be inferred:
    • Angles: The angles that once corresponded in triangle \( \triangle EFG \) will correspond to angles in \( \triangle XYZ \) (due to the property of rotation preserving angle measures).
    • Sides: The lengths of the sides will remain congruent since rotation preserves lengths.

Now let's evaluate the statements provided:

  1. \( \angle F \cong \angle Z \): True. After the rotation, angle F will correspond to angle Z.
  2. \( \overline{EG} \cong \overline{XY} \): True. Side EG corresponds to side XY.
  3. \( \angle E \cong \angle X \): False. Angle E corresponds to angle Y after rotation.
  4. \( \overline{EF} \cong \overline{XZ} \): False. Side EF corresponds to side YZ after rotation.

In summary, the statements that must be true are:

  • \( \angle F \cong \angle Z \)
  • \( \overline{EG} \cong \overline{XY} \)

Thus, both statements 1 and 2 are true.

1one answetr only

The statement that must be true is:

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