Question

prove that $\triangle dab \cong \triangle bcd$.
image of a quadrilateral with vertices a, b, c, d, and diagonal bd. ab and cd are marked as parallel (with arrows), and angles at a and c are marked.

statement	reason
2. pick statement	they are lengths of the same segment.
3. $\overline{ab} \parallel \overline{cd}$	given
4. $\angle abd \cong \angle cdb$	when a transversal crosses parallel lines, pick type of angle pair angles are congruent.
5. $\triangle dab \cong \triangle bcd$	pick congruence criterion congruence

Explanation:

Step1: Determine Statement 2

The reason for statement 2 is "They are lengths of the same segment", so the segment must be common to both triangles. The common segment in $\triangle DAB$ and $\triangle BCD$ is $DB$, so the statement is $DB \cong DB$ (or $\overline{DB} \cong \overline{DB}$).

Step2: Determine Angle Pair for Step 4

Since $AB \parallel CD$ and $DB$ is the transversal, $\angle ABD$ and $\angle CDB$ are alternate interior angles. So the type of angle pair is "alternate interior".

Step3: Determine Congruence Criterion for Step 5

We have $\angle A \cong \angle C$ (given), $DB \cong DB$ (common side), and $\angle ABD \cong \angle CDB$ (alternate interior angles). This fits the Angle - Angle - Side (AAS) congruence criterion.

Answer:

1. Statement 2: $\boldsymbol{DB \cong DB}$ (or $\boldsymbol{\overline{DB} \cong \overline{DB}}$)
2. Angle Pair (Step 4): Alternate Interior Angles
3. Congruence Criterion (Step 5): AAS (Angle - Angle - Side)