Question

complete the statements to prove that $\overline{ab}\cong\overline{cd}$ and $\overline{bc}\cong\overline{ad}$.
statement
quadrilateral abcd is a parallelogram.
$\overline{ab}\parallel\overline{cd}$ and $\overline{bc}\parallel\overline{ad}$
draw $\overline{ac}$. $\overline{ac}$ is a transversal that intersects two pairs of parallel lines,
$\overline{ab}$ and $\overline{cd}$ and $\overline{ad}$ and $\overline{bc}$.
$\angle cab\cong\angle acd$ and $\angle bca\cong\angle cad$
$m\angle cab = m\angle acd$ and $m\angle bca = m\angle cad$
$\overline{ac}=\overline{ac}$
$\triangle abc\cong\triangle cda$
$\overline{ab}\cong\overline{cd}$ and $\overline{bc}\cong\overline{ad}$
reason
given
definition of a parallelogram
drawing a line segment
congruent angles have equal measures.
alternate interior angles theorem
alternate exterior angles theorem
angles are congruent.

Explanation:

Step1: Determine the reason for $\angle CAB\cong\angle ACD$ and $\angle BCA\cong\angle CAD$

Since $\overline{AB}\parallel\overline{CD}$ and $\overline{AC}$ is a transversal, by the Alternate Interior Angles Theorem, $\angle CAB\cong\angle ACD$. Also, since $\overline{AD}\parallel\overline{BC}$ and $\overline{AC}$ is a transversal, by the Alternate Interior Angles Theorem, $\angle BCA\cong\angle CAD$.

Step2: Determine the reason for $\overline{AC}=\overline{AC}$

By the Reflexive Property of Congruence (a segment is congruent to itself), $\overline{AC}=\overline{AC}$.

Step3: Determine the reason for $\triangle ABC\cong\triangle CDA$

We have two pairs of congruent angles ($\angle CAB\cong\angle ACD$ and $\angle BCA\cong\angle CAD$) and one pair of congruent sides ($\overline{AC}=\overline{AC}$). By the Angle - Side - Angle (ASA) congruence criterion, $\triangle ABC\cong\triangle CDA$.

Step4: Determine the reason for $\overline{AB}\cong\overline{CD}$ and $\overline{BC}\cong\overline{AD}$

Since $\triangle ABC\cong\triangle CDA$, by the Corresponding Parts of Congruent Triangles are Congruent (CPCTC) theorem, $\overline{AB}\cong\overline{CD}$ and $\overline{BC}\cong\overline{AD}$.

Answer:

For $\angle CAB\cong\angle ACD$ and $\angle BCA\cong\angle CAD$: Alternate Interior Angles Theorem.
For $\overline{AC}=\overline{AC}$: Reflexive Property of Congruence.
For $\triangle ABC\cong\triangle CDA$: ASA (Angle - Side - Angle) congruence criterion.
For $\overline{AB}\cong\overline{CD}$ and $\overline{BC}\cong\overline{AD}$: CPCTC (Corresponding Parts of Congruent Triangles are Congruent).