Answer:
Option 3 is correct
SSS congruency theorem
Step-by-step explanation:
In ΔADC and ΔCBA
[tex]\overline{AB} \cong \overline {CD}[/tex] [Side] [Given]
[tex]\overline{AD} \cong \overline {BC}[/tex] [Side] [Given]
Reflexive property states that any value is equal to itself
[tex]\overline{AC} \cong \overline {AC}[/tex] [Side] [Reflexive Property]
SSS -Side-Side-Side postulates states that that if three sides of one triangle are congruent to three sides of another triangle, then these two triangles are congruent.
Therefore,
ΔADC [tex]\cong[/tex] ΔCBA [By SSS]
By CPCT (Corresponding Part of Congruent Triangle]
[tex]\angle DAC \cong \angle BCA[/tex] and
[tex]\angle ACD \cong \angle CAB[/tex]
Alternate interior angle states that a pair of angles on the inner side of each of those two lines but on opposite sides of the transversal.
therefore, by definition of alternate interior angle ;
[tex]\angle DAC[/tex] and [tex]\angle BCA [/tex] are alternate interior angle
also, [tex]\angle ACD[/tex] and [tex]\angle CAB [/tex] are alternate interior angle
By converse of the alternate interior angle theorem states that if two lines are cut by a transversal and the alternate interior angles are congruent, then the lines are parallel.
therefore, we have
[tex]\overline{AB} || \overline{CD}[/tex] ;
[tex]\overline{AD} || \overline{BC}[/tex]
Then,by the definition of parallelogram that a four sided flat shape with straight sides where opposite sides are parallel.
⇒ABCD is parallelogram hence proved!
