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

According to the given information, AB || DC and BC || AD. Construct a diagonal from A to C with a straightedge. Angles BAC and DCA are congruent by the Alternate Interior Angles Theorem. Angles BCA and DAC are congruent by the Alternate Interior Angles Theorem. 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 BAC and DCA are congruent by the Same-Side Interior Angles Theorem.
B. Diagonal BD is congruent to itself by the Reflexive Property of Equality.
C. Diagonal AC is congruent to itself by the Reflexive Property of Equality.
D. Angles ABC and CDA are congruent according to a property of parallelograms (opposite angles congruent).

Answer :

Final answer:

Option C, 'Diagonal AC is congruent to itself by the Reflexive Property of Equality', accurately completes the proof completion that the opposite sides of a parallelogram are congruent.The correct option to complete the proof is: D)

Explanation:

The sentence that accurately completes the proof is: C) 'Diagonal AC is congruent to itself by the Reflexive Property of Equality'. The proof has already established that triangles BCA and DAC are congruent according to the Angle-Side-Angle (ASA) Theorem.

Thus, by the Corresponding Parts of Congruent Triangles are Congruent (CPCTC) principle, we know that opposite sides AB and CD, as well as sides BC and DA, are congruent.

To complete the proof, we need to establish that Diagonal AC is congruent to itself, which is proven by utilizing the Reflexive Property of Equality, stating that any side, line or angle is congruent to itself.The correct option to complete the proof is: D)

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Rewritten by : Jeany

Final answer:

The best sentence to complete the proof is 'Diagonal AC is congruent to itself by the Reflexive Property of Equality.' This is because it brings the necessary relation between triangles to apply the ASA Theorem and conclude the congruence of opposite sides using CPCTC.

Explanation:

To complete the proof that the opposite sides of parallelogram ABCD are congruent, the missing statement is option (C): 'Diagonal AC is congruent to itself by the Reflexive Property of Equality.' The reason is that BCA and DAC are congruent by the Alternate Interior Angles Theorem and Angle-Side-Angle (ASA) Theorem. However, to complete the proof, an added connectivity is required, this is provided by the diagonal AC which is congruent to itself. Thus, invoking the Reflexive Property of Equality completes the congruence correspondence between triangles BCA and DAC, hence, by Corresponding Parts of Congruent Triangles are Congruent (CPCTC), we can conclude that opposite sides AB and CD and sides BC and DA are congruent.

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