Question

given this proof, what is the missing step?
given: \\( \overline { a d } \cong \overline { d c } \\) and \\( \angle a \cong \angle c \\)
prove: \\( \triangle a b d \cong \triangle c b d \\)
\\( \overline { b d } \cong \overline { b d } \\), reflexive property
\\( \angle a d b \cong \angle c d b \\), perpendiculars form congruent right angles.
\\( \angle a b d \cong \angle c b d \\), if two pairs of angles are congruent, the third pair of angles are congruent.
\\( \overline { b c } \cong \overline { b a } \\), the sides opposite congruent angles of a triangle are congruent.

Explanation:

Step1: Recall the SAS Congruence Theorem

The SAS (Side - Angle - Side) Congruence 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 information and the required elements for SAS

We are given \( \overline{AD}\cong\overline{DC}\) (a side) and \( \angle A\cong\angle C\) (an angle). For the SAS theorem, we need another pair of sides. The side \( \overline{BD}\) is common to both \( \triangle ABD\) and \( \triangle CBD\). By the reflexive property of congruence, any segment is congruent to itself. So, \( \overline{BD}\cong\overline{BD}\).

Answer:

A. \( \overline{BD}\cong\overline{BD}\), Reflexive Property