Question

given: $overline{ab}paralleloverline{cd}$ and $overline{ad}paralleloverline{bc}$
prove: $overline{ab}congoverline{cd}$ and $overline{ad}congoverline{bc}$
complete the flowchart proof.
asa
reflexive property of congruence
cpctc
alternate interior angles theorem
alternate exterior angles theorem
sas
construct $overline{ac}$
two points define a line
$overline{ab}paralleloverline{cd}$
$overline{ad}paralleloverline{bc}$
given
$overline{ac}congoverline{ac}$
$angle daccongangle bca$
$angle acdcongangle bac$
$\triangle acdcong\triangle cab$
$overline{ab}congoverline{cd}$
$overline{ad}congoverline{bc}$

Explanation:

Step1: Construct diagonal AC

Two - points define a line, so we can construct $\overline{AC}$.

Step2: Identify alternate interior angles

Since $\overline{AB}\parallel\overline{CD}$ and $\overline{AD}\parallel\overline{BC}$, by the alternate - interior angles theorem, $\angle DAC\cong\angle BCA$ and $\angle ACD\cong\angle BAC$.

Step3: Use the reflexive property

$\overline{AC}\cong\overline{AC}$ by the reflexive property of congruence.

Step4: Prove triangle congruence

By the ASA (Angle - Side - Angle) congruence criterion, $\triangle ACD\cong\triangle CAB$ because we have two pairs of congruent angles and a common side.

Step5: Use CPCTC

Corresponding Parts of Congruent Triangles are Congruent (CPCTC). So, $\overline{AB}\cong\overline{CD}$ and $\overline{AD}\cong\overline{BC}$.

Answer:

The flowchart should have the following labels in order from left - to - right and top - to - bottom: "construct $\overline{AC}$, two points define a line", "alternate interior angles theorem", "reflexive property of congruence", "ASA", "CPCTC".