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If you wanted to explain why ABD is congruent to CBD by the SAS Congruence Theorem, which of the following would be helpful to know?

BD is the angle Bisector of BCD
AC is the angle bisector of BAD
AC is the angle bisector of BCD
BC is the angle bisector of ABC

1 answer

To explain why triangles ABD and CBD are congruent by the Side-Angle-Side (SAS) Congruence Theorem, you need to establish two sides and the included angle between them as congruent.

The information that would be most helpful to know is:

BD is the angle bisector of BCD.

Here's why this is helpful:

  • If BD is the angle bisector of angle BCD, then angle ABD is congruent to angle CBD (since an angle bisector divides an angle into two equal parts).
  • You can use the shared side BD between triangles ABD and CBD as one of the congruent sides.
  • Therefore, you would have two sides (AB and CB if we consider them equal or any lengths provided) and the included angle (angle ABD and CBD being equal), which would confirm congruence by the SAS theorem.

The other options concerning AC being the angle bisector of either BAD or BCD, or BC being the angle bisector, do not directly help in establishing the congruence between the specified triangles ABD and CBD as clearly as identifying BD as the angle bisector of BCD does.