Question

what additional information is needed to prove that (\triangle abccong\triangle dbf) using the sas congruence theorem?
( \bigcirc ) the measure of ( angle c ) and ( angle d )
( \bigcirc ) the side lengths of ( ac ) and ( df )
( \bigcirc ) the measure of ( angle f ) and ( angle a )
( \bigcirc ) the side lengths of ( ab ) and ( db )

Explanation:

The SAS (Side - Angle - Side) congruence theorem requires two sides and the included angle of one triangle to be equal to the corresponding two sides and included angle of another triangle. In \(\triangle ABC\) and \(\triangle DBF\), we already know that \(\angle ABC=\angle DBF\) (vertically opposite angles are equal). To apply SAS, we need the lengths of the sides adjacent to the equal angles. The sides adjacent to \(\angle ABC\) in \(\triangle ABC\) is \(AB\) and \(BC\), and the sides adjacent to \(\angle DBF\) in \(\triangle DBF\) is \(DB\) and \(BF\). Since \(BC = BF\) (marked as equal in the figure), we need the lengths of \(AB\) and \(DB\).

Answer:

the side lengths of \(AB\) and \(DB\)