Question

triangle abc undergoes a series of transformations to make triangle abc.
which statements are true?
select all that apply.
a a rotation of triangle abc by 60°, followed by a reflection over the line x = 3, will result in triangle abc being congruent to triangle abc.
b a reflection of triangle abc over the y - axis, followed by a dilation by a factor of 2, will result in triangle abc being congruent to triangle abc.
c a rotation of triangle abc by 180°, followed by a translation of 3 units horizontally, will result in triangle abc not being congruent to triangle abc.
d a dilation of triangle abc by a factor of 0.5, followed by a translation of 5 units vertically, will result in triangle abc not being congruent to triangle abc.
part of option e of triangle abc of 3 units horizontally and 2 units vertically, followed by a reflection over the x - axis, will result in triangle abc being congruent to triangle abc

Explanation:

Step1: Recall congruence rules

Congruent figures have identical size/shape. Rigid transformations (rotation, reflection, translation) preserve congruence; dilations (non-rigid) do not.

Step2: Analyze Option A

Rotation + reflection are rigid.
$\text{Congruence preserved: } \triangle ABC \cong \triangle A'B'C'$ → True.

Step3: Analyze Option B

Reflection (rigid) + dilation (non-rigid, scale=2).
$\text{Size changes: } \triangle ABC
cong \triangle A'B'C'$ → False.

Step4: Analyze Option C

Rotation + translation are rigid.
$\text{Congruence preserved: } \triangle ABC \cong \triangle A'B'C'$ → Statement claims no congruence, so False.

Step5: Analyze Option D

Dilation (non-rigid, scale=0.5) + translation.
$\text{Size changes: } \triangle ABC
cong \triangle A'B'C'$ → True.

Step6: Analyze Option E

Translation + reflection are rigid.
$\text{Congruence preserved: } \triangle ABC \cong \triangle A'B'C'$ → True.

Answer:

A. A rotation of triangle ABC by 60°, followed by a reflection over the line x = 3, will result in triangle ABC being congruent to triangle A'B'C'
D. A dilation of triangle ABC by a factor of 0.5, followed by a translation of 5 units vertically, will result in triangle ABC not being congruent to triangle A'B'C'
E. A translation of triangle ABC of 3 units horizontally and 2 units vertically, followed by a reflection over the x-axis, will result in triangle ABC being congruent to triangle A'B'C'