Question

using transformations to prove congruence
is there a rigid transformation that maps triangle abc to
triangle abd? if so, which transformation?
○ yes, because a translation to the right will map δabc
to δabd
○ yes, because a rotation about point b will map δabc
to δabd
○ yes, because a reflection across \\(\overline{ba}\\) will map δabc to
δabd
○ no, because no rigid transformation will map δabc to
δabd

Explanation:

To determine the rigid transformation mapping \( \triangle ABC \) to \( \triangle ABD \), we analyze each option:

• Translation: A translation moves the figure without rotation/reflection. The triangles are symmetric over \( \overline{BA} \), not a translation. Eliminate this.
• Rotation about \( B \): Rotation would change the orientation relative to \( BA \), but the triangles are symmetric across \( BA \), not a rotation. Eliminate this.
• Reflection across \( \overline{BA} \): Reflecting \( \triangle ABC \) over \( \overline{BA} \) (the common side) would map \( C \) to \( D \), since \( \angle ABC \) and \( \angle ABD \) are equal (marked angles) and \( AB \) is shared. This matches the symmetry.
• No rigid transformation: Incorrect, as reflection across \( \overline{BA} \) works.

Thus, the correct option is the one describing reflection across \( \overline{BA} \).

Answer:

yes, because a reflection across \(\overline{BA}\) will map \(\triangle ABC\) to \(\triangle ABD\)