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 share side \( BA \), so translation is incorrect (eliminates first option).
• Rotation: Rotating about point \( B \) would not align \( C \) to \( D \) as the triangles are symmetric over \( \overline{BA} \), not rotated (eliminates second option).
• Reflection: Reflecting \( \triangle ABC \) across \( \overline{BA} \) (the common side) will map \( C \) to \( D \), as the triangles are mirror images over \( \overline{BA} \). This matches the symmetry shown (angles at \( B \) and \( A \) are equal, and \( BA \) is a shared side).
• No transformation: The triangles are congruent (symmetric), so a rigid transformation exists (eliminates fourth option).

Answer:

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