Question

decide if each statement describing figures wxyz and wxyz after a 180° clockwise rotation is true or false. image of coordinate grid with quadrilateral wxyz the x - coordinate of z is the same as the x - coordinate of z. the coordinates of the vertices of wxyz will be the same if the rotation is 180° counterclockwise. xy = xy the y - coordinate of w is the opposite of the y - coordinate of w. the rotation does not preserve congruence because the image is has a different orientation than the original figure. m∠z = m∠z

Explanation:

To solve this, we use the properties of a \(180^\circ\) rotation (clockwise or counterclockwise, as they are equivalent for \(180^\circ\)): A \(180^\circ\) rotation about the origin transforms a point \((x, y)\) to \((-x, -y)\). Rotations preserve distance (lengths) and angle measures (congruence is preserved).

1. The \(x\)-coordinate of \(Z'\) is the same as the \(x\)-coordinate of \(Z\).

For a \(180^\circ\) rotation, \(Z(x, y) \to Z'(-x, -y)\). The \(x\)-coordinate of \(Z'\) is \(-x\) (opposite of \(Z\)’s \(x\)-coordinate). Thus, this statement is False.

2. The coordinates of the vertices of \(W'X'Y'Z'\) will be the same if the rotation is \(180^\circ\) counterclockwise.

A \(180^\circ\) clockwise rotation and a \(180^\circ\) counterclockwise rotation are identical (both map \((x, y) \to (-x, -y)\)). Thus, the coordinates will be the same. This statement is True.

3. \(XY' = XY\)

Rotations preserve distance (length of segments). So the length of \(XY\) (original) equals the length of \(XY'\) (after rotation). This statement is True.

4. The \(y\)-coordinate of \(W'\) is the opposite of the \(y\)-coordinate of \(W\).

For \(W(x, y) \to W'(-x, -y)\), the \(y\)-coordinate of \(W'\) is \(-y\) (opposite of \(W\)’s \(y\)-coordinate). This statement is True.

5. The rotation does not preserve congruence because the image has a different orientation than the original figure.

Rotations preserve congruence (lengths, angles, and shape are unchanged). Orientation (direction) may change, but congruence is still preserved. Thus, this statement is False.

6. \(m\angle Z' = m\angle Z\)

Rotations preserve angle measures (congruence of angles). So the measure of \(\angle Z'\) equals the measure of \(\angle Z\). This statement is True.

Final Answers (matching each statement to True/False):

1. False
2. True
3. True
4. True
5. False
6. True

Answer:

To solve this, we use the properties of a \(180^\circ\) rotation (clockwise or counterclockwise, as they are equivalent for \(180^\circ\)): A \(180^\circ\) rotation about the origin transforms a point \((x, y)\) to \((-x, -y)\). Rotations preserve distance (lengths) and angle measures (congruence is preserved).

1. The \(x\)-coordinate of \(Z'\) is the same as the \(x\)-coordinate of \(Z\).

For a \(180^\circ\) rotation, \(Z(x, y) \to Z'(-x, -y)\). The \(x\)-coordinate of \(Z'\) is \(-x\) (opposite of \(Z\)’s \(x\)-coordinate). Thus, this statement is False.

2. The coordinates of the vertices of \(W'X'Y'Z'\) will be the same if the rotation is \(180^\circ\) counterclockwise.

A \(180^\circ\) clockwise rotation and a \(180^\circ\) counterclockwise rotation are identical (both map \((x, y) \to (-x, -y)\)). Thus, the coordinates will be the same. This statement is True.

3. \(XY' = XY\)

Rotations preserve distance (length of segments). So the length of \(XY\) (original) equals the length of \(XY'\) (after rotation). This statement is True.

4. The \(y\)-coordinate of \(W'\) is the opposite of the \(y\)-coordinate of \(W\).

For \(W(x, y) \to W'(-x, -y)\), the \(y\)-coordinate of \(W'\) is \(-y\) (opposite of \(W\)’s \(y\)-coordinate). This statement is True.

5. The rotation does not preserve congruence because the image has a different orientation than the original figure.

Rotations preserve congruence (lengths, angles, and shape are unchanged). Orientation (direction) may change, but congruence is still preserved. Thus, this statement is False.

6. \(m\angle Z' = m\angle Z\)

Rotations preserve angle measures (congruence of angles). So the measure of \(\angle Z'\) equals the measure of \(\angle Z\). This statement is True.

Final Answers (matching each statement to True/False):

1. False
2. True
3. True
4. True
5. False
6. True