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 BCA and DAC are congruent by the same reasoning. Triangles BCA and DAC are congruent according to the Angle-Side-Angle (ASA) Theorem. By CPCTC, opposite sides AB and CD, as well as sides BC and DA, are congruent.
Which sentence accurately completes the proof?

A) Angles ABC and CDA are corresponding parts of congruent triangles, which are congruent (CPCTC).

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

C) Angles BAC and DCA are congruent by the Same-Side Interior Angles Theorem.

D) Angles BAC and DCA are congruent by the Alternate Interior Angles Theorem.

Answer 1

D. Angles BAC and DCA are congruent by the Alternate interior angles theorem.

Explanation:

Given ABCD is a parallelogram.

AB=CD and BC=AD

`AB\parallel CD \; and\; BC\parallel AD`

To prove that
`AB\cong CD\; and\; BC\cong AD`.

Proof: According to given information

`AB\parallel CD` and segment
`BC\parallel AD`.

Construct diagonal AC with a straightedge.

In
`\triangle ABC\; and\; \triangle ADC`

AC=AC

Reflexive property of equality.

`m\angle BCA\cong m\angle DAC`

By alternate interior angles theorem.

`m\angle BAC\cong m\angle DCA`

By alternate interior angles theorem.

`\triangle BCA\cong\triangle DAC`

By ASA( Angle-Side_Angle) theorem.

By CPCTC opposite sides AB and CD, as well as BC and DA are congruent.

Hence proved.

Answer 2

D) Angles BAC and DCA are congruent by the Alternate Interior Angles Theorem.