IIf <ABC is mapped to < A' B' C' using a rigid motion, which of the following does not need to be true?

Question

AB is parellal to A' B'
BC=B'C'
the midpoints are mapped
m<ABC=m<A' B' C'

Answer 1

To determine which statement does not need to be true when triangle \( \triangle ABC \) is mapped to triangle \( \triangle A'B'C' \) using a rigid motion, let's analyze each option:

1. \( AB \) is parallel to \( A'B' \): This is true because rigid motions preserve the orientation and the relative orientation of sides. Therefore, corresponding sides of congruent triangles after a rigid motion remain parallel.

2. \( BC = B'C' \): This must be true because rigid motions (like translations, rotations, and reflections) preserve distances. Hence, the lengths of corresponding sides will remain equal.

3. The midpoints are mapped: This does not necessarily need to be true. Rigid motions do preserve the orientation and lengths of segments, but they do not necessarily map midpoints to midpoints unless the motion is specifically a translation. For rotations or reflections, the midpoints of segments may not correspond.

4. \( m\angle ABC = m\angle A'B'C' \): This is true as rigid motions preserve angles.

Thus, the statement that does not need to be true among the options provided is the midpoints are mapped.