Question

given quadrilateral abcd with (overline{ad} cong overline{bc}), (overline{ab} cong overline{dc})
prove: (abcd) is a parallelogram
move statements and reasons to the table to complete the proof.

statements	reasons
	reflexive property
(angle dac cong angle bca), (angle dca cong angle bac)
	converse of alternate interior angles theorem
(abcd) is a parallelogram	definition of parallelogram

(overline{ac} cong overline{ac})
(\triangle dac cong \triangle bca)
(da parallel bc), (ab parallel dc)
cpctc
asa
sss
(\triangle dac cong \triangle bcd)
sas

Explanation:

Step1: Identify reflexive side

$\overline{AC} \cong \overline{AC}$

Step2: Prove triangle congruence

$\triangle DAC \cong \triangle BCA$ (SSS)

Step3: Justify angle congruence

CPCTC

Step4: Prove parallel sides

$\overline{AD} \parallel \overline{BC}, \overline{AB} \parallel \overline{DC}$

Answer:

Completed proof table:

Statements	Reasons
$\overline{AC} \cong \overline{AC}$	Reflexive Property
$\triangle DAC \cong \triangle BCA$	SSS
$\angle DAC \cong \angle BCA, \angle DCA \cong \angle BAC$	CPCTC
$\overline{AD} \parallel \overline{BC}, \overline{AB} \parallel \overline{DC}$	Converse of Alternate Interior Angles Theorem
$ABCD$ is a parallelogram	Definition of parallelogram