Question

segment \\(\overline{ac}\\) is the angle bisector of \\(\angle bad\\). point \\(c\\) is collinear with points \\(b\\) and \\(d\\).
the figure is not to scale.

figure of points a, d, c, b with lines ad, ac, ab, dc, cb

which of the following additional statements would allow us to prove that \\(\angle adb \cong \angle abd\\)?
choose 1 answer:
\\(\boldsymbol{\circ}\\) \\(\overline{da} \cong \overline{ba}\\) only
\\(\boldsymbol{\circ}\\) \\(\overline{da} \cong \overline{bd}\\) only
\\(\boldsymbol{\circ}\\) either statement is sufficient.
\\(\boldsymbol{\circ}\\) we need both statements.
\\(\boldsymbol{\circ}\\) even with both statements, we still could not prove that \\(\angle adb \cong \angle abd\\).

Explanation:

To prove \(\angle ADB \cong \angle ABD\), we can use the Isosceles Triangle Theorem (if two sides of a triangle are congruent, the angles opposite them are congruent). In \(\triangle ABD\), if \(\overline{DA} \cong \overline{BA}\), then the angles opposite these sides (\(\angle ABD\) and \(\angle ADB\) respectively) would be congruent. Let's analyze the options:

• Option A: If \(\overline{DA} \cong \overline{BA}\), in \(\triangle ABD\), by the Isosceles Triangle Theorem, \(\angle ADB \cong \angle ABD\) (since \(\angle ADB\) is opposite \(\overline{BA}\) and \(\angle ABD\) is opposite \(\overline{DA}\)).
• Option B: \(\overline{DA} \cong \overline{BD}\) would relate to angles \(\angle ABD\) and \(\angle BAD\) (by Isosceles Triangle Theorem), not directly to \(\angle ADB \cong \angle ABD\).
• Option C: Since only Option A works, this is incorrect.
• Option D: We don't need both, so this is incorrect.
• Option E: We can prove it with Option A, so this is incorrect.

So only \(\overline{DA} \cong \overline{BA}\) (Option A) is sufficient.

Answer:

A. \(\overline{DA} \cong \overline{BA}\) only