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)

Explanation:

We have to prove that the opposite sides of parallelogram ABCD are congruent.

Proof: 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.

From ΔABC and ΔACD, we have

∠BAC≅∠DCA (Alternate interior angles theorem)

AC≅AC (Reflexive property)

∠BCA≅∠DAC(Alternate interior angles theorem)

Therefore, ΔABC ≅ ΔACD by ASA rule of congruency.

Thus, option B is correct that is Triangles BCA and DAC are congruent according to the Angle-Side-Angle (ASA) Theorem.

Answer 2

The correct option is B.

Explanation:

Given information:
`AB\parallel DC` and
`BC\parallel AD`.

Draw a diagonal AC.

In triangle BCA and DAC,

`AC\cong AC` (Reflexive Property of Equality)

`\angle BAC\cong \angle DCA` ( Alternate Interior Angles Theorem)

`\angle BCA\cong \angle DAC` ( Alternate Interior Angles Theorem)

The ASA (Angle-Side-Angle) postulate states that two triangles are congruent if two corresponding angles and the included side of are congruent.

By ASA postulate,

`\triangle BCA\cong \triangle DAC`

Therefore option B is correct.