Question

△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¯¯¯¯¯¯¯¯

Answer 1

To analyze the effect of rotating triangle ΔEFG by -90 degrees (clockwise) to create triangle ΔXYZ, we can map the points as follows:

When you rotate a point (x, y) by -90 degrees, the new coordinates (x', y') will be:

- x' = y
- y' = -x

Applying this transformation to the points E, F, and G will result in the points X, Y, and Z.

Let’s assume:
- E corresponds to (x₁, y₁)
- F corresponds to (x₂, y₂)
- G corresponds to (x₃, y₃)

Then after rotation:
- X will be (y₁, -x₁)
- Y will be (y₂, -x₂)
- Z will be (y₃, -x₃)

Now we can evaluate the statements one by one:

1. **∠F ≅ ∠Z**: This is true because angle F in ΔEFG maps to angle Z in ΔXYZ after the rotation. Since the shape and size of the triangle do not change in a rotation, the angles corresponding under this transformation will be congruent.

2. **EG ≅ XY**: This is false. The sides of the triangle do not become congruent when taking points from triangle EFG to triangle XYZ since a rotation does not change side lengths.

3. **∠E ≅ ∠X**: This is true as well. Angle E in ΔEFG corresponds to angle X in ΔXYZ after the rotation.

4. **EF ≅ XZ**: This is also true. The side lengths of the triangle remain the same despite the rotation, so side EF in ΔEFG corresponds to side XZ in ΔXYZ.

The statements that are true must include:
- **∠F ≅ ∠Z** (True)
- **∠E ≅ ∠X** (True)
- **EF ≅ XZ** (True)

**Final Verdict**:
The statements **∠F ≅ ∠Z**, **∠E ≅ ∠X**, and **EF ≅ XZ** must be true after the triangle is rotated. The statement **EG ≅ XY** is false.