Question

1. this two-column proof shows that $overleftrightarrow{ac}\parallel\overleftrightarrow{bd}$ in the attached diagram, but it is incomplete.

to prove: $overleftrightarrow{ac}\parallel\overleftrightarrow{bd}$

statements	reasons
$\angle abd\cong\angle dca$	given
________	________
$\angle dca$ and $\angle bdc$ are supplementary	substitution
$overleftrightarrow{ac}\parallel\overleftrightarrow{bd}$	consecutive interior angles converse

Explanation:

Step1: Identify consecutive interior angles

Since $\overleftrightarrow{AB} \parallel \overleftrightarrow{CD}$, $\angle ABD$ and $\angle BDC$ are consecutive interior angles, so they are supplementary:
$\angle ABD + \angle BDC = 180^\circ$

Step2: Substitute congruent angles

We know $\angle ABD \cong \angle DCA$, so replace $\angle ABD$ with $\angle DCA$:
$\angle DCA + \angle BDC = 180^\circ$

Answer:

Statements	Reasons
$\angle ABD \cong \angle DCA$	Given
$\angle ABD$ and $\angle BDC$ are supplementary	Consecutive interior angles theorem
$\angle DCA$ and $\angle BDC$ are supplementary	Substitution
$\overleftrightarrow{AC} \parallel \overleftrightarrow{BD}$	Consecutive interior angles converse

The missing statement is: $\boldsymbol{\angle ABD \text{ and } \angle BDC \text{ are supplementary}}$
The missing reason is: $\boldsymbol{\text{Consecutive interior angles theorem}}$