Question

determine whether each statement about the triangles is true. choose true or false from each drop-down menu for each statement.

△abc is similar to △abc.
△abc is congruent to △abc.
△abc can be reflected across y-axis to form △abc, and then △abc can be rotated 90° clock about the origin to form △abc
△abc can be rotated 90° counterclockwise about the origin to form △abc, and then △abc can be dilated by a scale factor of 2 with the center of dilation at the origin to form △abc.

Explanation:

To solve this, we analyze each statement about triangle transformations (similarity, congruence, and transformations like reflection, rotation, dilation):

Statement 1: $\boldsymbol{\triangle ABC \text{ is similar to } \triangle A''B''C''}$

Similarity is preserved under rigid transformations (reflection, rotation) and dilation (scaling). If $\triangle A''B''C''$ is formed by rigid transformations or dilation from $\triangle ABC$, their corresponding angles are equal, and sides are proportional. Thus, this is True.

Statement 2: $\boldsymbol{\triangle ABC \text{ is congruent to } \triangle A''B''C''}$

Congruence requires identical shape and size (rigid transformations only: reflection, rotation, translation—no dilation). If $\triangle A''B''C''$ involves dilation (scaling), size changes, so congruence fails. Thus, this is False.

Statement 3: $\boldsymbol{\triangle ABC \text{ reflected over } y\text{-axis} \to \triangle A'B'C', \text{ then rotated } 90^\circ \text{ clockwise} \to \triangle A''B''C''}$

Reflection over $y$-axis and rotation $90^\circ$ clockwise are rigid transformations (preserve size/shape). If the sequence maps $\triangle ABC$ to $\triangle A''B''C''$, this is True (assuming the transformation sequence is valid).

Statement 4: $\boldsymbol{\triangle ABC \text{ rotated } 90^\circ \text{ counterclockwise} \to \triangle A'B'C', \text{ then dilated by scale factor } 2 \to \triangle A''B''C''}$

Rotation is rigid (preserves size/shape), but dilation by 2 changes size. If $\triangle A''B''C''$ is a dilation of $\triangle A'B'C'$, it is similar but not congruent. However, the question asks if the transformation sequence is valid. If the dilation is part of forming $\triangle A''B''C''$, and the rotation + dilation sequence is described correctly, we check:

• Rotation $90^\circ$ counterclockwise: rigid (preserves size/shape).
• Dilation by 2: scales size.

If the problem implies this sequence forms $\triangle A''B''C''$, and the dilation is intended, this is True (if the dilation is part of the transformation).

Final Answers (for each dropdown):

1. $\triangle ABC$ is similar to $\triangle A''B''C''$: $\boldsymbol{\text{True}}$
2. $\triangle ABC$ is congruent to $\triangle A''B''C''$: $\boldsymbol{\text{False}}$
3. Reflection + rotation sequence: $\boldsymbol{\text{True}}$ (if the transformation is valid)
4. Rotation + dilation sequence: $\boldsymbol{\text{True}}$ (if the transformation is valid)

(Note: For the exact answer, confirm the transformation details, but based on typical problems, the congruence statement is False, and similarity/valid transformation sequences are True.)

Answer:

To solve this, we analyze each statement about triangle transformations (similarity, congruence, and transformations like reflection, rotation, dilation):

Statement 1: $\boldsymbol{\triangle ABC \text{ is similar to } \triangle A''B''C''}$

Similarity is preserved under rigid transformations (reflection, rotation) and dilation (scaling). If $\triangle A''B''C''$ is formed by rigid transformations or dilation from $\triangle ABC$, their corresponding angles are equal, and sides are proportional. Thus, this is True.

Statement 2: $\boldsymbol{\triangle ABC \text{ is congruent to } \triangle A''B''C''}$

Congruence requires identical shape and size (rigid transformations only: reflection, rotation, translation—no dilation). If $\triangle A''B''C''$ involves dilation (scaling), size changes, so congruence fails. Thus, this is False.

Statement 3: $\boldsymbol{\triangle ABC \text{ reflected over } y\text{-axis} \to \triangle A'B'C', \text{ then rotated } 90^\circ \text{ clockwise} \to \triangle A''B''C''}$

Reflection over $y$-axis and rotation $90^\circ$ clockwise are rigid transformations (preserve size/shape). If the sequence maps $\triangle ABC$ to $\triangle A''B''C''$, this is True (assuming the transformation sequence is valid).

Statement 4: $\boldsymbol{\triangle ABC \text{ rotated } 90^\circ \text{ counterclockwise} \to \triangle A'B'C', \text{ then dilated by scale factor } 2 \to \triangle A''B''C''}$

Rotation is rigid (preserves size/shape), but dilation by 2 changes size. If $\triangle A''B''C''$ is a dilation of $\triangle A'B'C'$, it is similar but not congruent. However, the question asks if the transformation sequence is valid. If the dilation is part of forming $\triangle A''B''C''$, and the rotation + dilation sequence is described correctly, we check:

• Rotation $90^\circ$ counterclockwise: rigid (preserves size/shape).
• Dilation by 2: scales size.

If the problem implies this sequence forms $\triangle A''B''C''$, and the dilation is intended, this is True (if the dilation is part of the transformation).

Final Answers (for each dropdown):

1. $\triangle ABC$ is similar to $\triangle A''B''C''$: $\boldsymbol{\text{True}}$
2. $\triangle ABC$ is congruent to $\triangle A''B''C''$: $\boldsymbol{\text{False}}$
3. Reflection + rotation sequence: $\boldsymbol{\text{True}}$ (if the transformation is valid)
4. Rotation + dilation sequence: $\boldsymbol{\text{True}}$ (if the transformation is valid)

(Note: For the exact answer, confirm the transformation details, but based on typical problems, the congruence statement is False, and similarity/valid transformation sequences are True.)