Question

The following is an incomplete paragraph proving that the opposite sides of parallelogram ABCD are congruent:

Parallelogram ABCD is shown where segment AB is parallel to segment DC and segment BC is parallel to segment AD.

According to the given information, segment AB is parallel to segment DC and segment BC is parallel to segment AD.. Construct diagonal A C with a straightedge. It is congruent to itself by the Reflexive Property of Equality. Angles BAC and DCA are congruent by the Alternate Interior Angles Theorem. Angles BCA and DAC are congruent by the same theorem. __________. By CPCTC, opposite sides AB and CD, as well as sides BC and DA, are congruent.

Which sentence accurately completes the proof?

Triangles BCA and DAC are congruent according to the Angle-Angle-Side (AAS) Theorem.
Triangles BCA and DAC are congruent according to the Angle-Side-Angle (ASA) Theorem.
Angles ABC and CDA are congruent according to a property of parallelograms (opposite angles congruent).
Angles BAD and ADC, as well as angles DCB and CBA, are supplementary by the Same-Side Interior Angles Theorem.

Answer 1

Answer:  Triangles BCA and DAC are congruent according to the Angle-Side-Angle (ASA) Theorem.

Step-by-step explanation:

Given : ABCD is a parallelogram.

That is, AB ║ CD and AD ║BC

We have to prove that: AB≅CD and AD≅BC

Proof:

Construct diagonal AC in the parallelogram ABCD.

Since, AC ≅ AC ( reflexive)

∠ BAC ≅ ∠ DCA  ( By the alternative interior angle theorem)

∠ BCA ≅ ∠ DAC   ( By the alternative interior angle theorem)

⇒ Δ BCA ≅ Δ DAC ( By ASA congruence postulate )

⇒ AB≅CD as well as AD≅BC ( BY CPCTC )

Thus, the opposite side of the parallelogram are congruent.

Answer 2

Diagonal AC is congruent to itself by the Reflexive Property of Equality.