Question

can \\( \triangle a b d \\) be proved congruent to \\( \triangle c d b \\) by the side-angle-side triangle congruency theorem from the figure?
if yes, select the appropriate pairs of corresponding parts to show that the triangles are congruent using the side-angle-side triangle congruency theorem. if no, select \cannot be proved.\
\\( \square \angle b c d \cong \angle b d a \\)
\\( \square b d \cong d b \\)
\\( \square a d \cong c b \\)
\\( \square \angle a b d \cong \angle c b d \\)
\\( \square \angle a d b \cong \angle c b d \\)
\\( \square a b \cong b d \\)
\\( \square \\) cannot be proved

Explanation:

Step1: Recall the Side - Angle - Side (SAS) Congruence Theorem

The SAS theorem states that if two sides and the included angle of one triangle are congruent to two sides and the included angle of another triangle, then the two triangles are congruent.

Step2: Analyze the given figure

From the figure, we can observe that \(AD\cong CB\) (given as marked in the figure). \(BD\) is a common side, so \(BD\cong DB\) (reflexive property of congruence). Also, since \(AD\parallel CB\) (the arrows indicating parallelism), \(\angle ADB\cong\angle CBD\) (alternate interior angles).

Step3: Check the pairs for SAS

For \(\triangle ABD\) and \(\triangle CDB\):

• Side: \(AD\cong CB\)
• Angle: \(\angle ADB\cong\angle CBD\)
• Side: \(BD\cong DB\)

Answer:

\(\boxed{BD\cong DB}\), \(\boxed{AD\cong CB}\), \(\boxed{\angle ADB\cong\angle CBD}\)