Question

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

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?

A) Triangles BCA and DAC are congruent according to the Angle-Angle-Side (AAS) Theorem.

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

C) Angles ABC and CDA are congruent according to a property of parallelograms (opposite angles congruent).

D) Angles BAD and ADC, as well as angles DCB and CBA, are supplementary by the Same-Side Interior Angles Theorem.

Answer 1

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

Explanation:

The Angle-Side-Angle (ASA) Theorem states that if two angles and the included side of one triangle are congruent to two angles and the included side of another triangle, then the triangles are congruent. (The included side is the side between the vertices of the two angles.)

In this case:

• ∠BAC ≅ ∠DCA

• ∠BCA ≅ ∠DAC

• the included congruent side is diagonal AC

Answer 2

(B) Triangles BCA and DAC are congruent according to the angle-side-angle (ASA) theorem.

Explanation:

In order to prove opposite side of parallelogram are congruent, we take

In ΔBCA and ΔDAC, we have

∠BAC=∠DCA( alternate angles)

AC=AC( reflexive property)

∠BCA=∠DAC( alternate angles)

Therefore, by ASA rule of congruency,

ΔBCA ≅ ΔDAC.

Bu CPCTC, AB and CD and BC and DA are congruent.

Hence, option B accurately completes the proof.