In the diagram below, quadrilateral ABCD…

Maths Questions

In the diagram below, quadrilateral ABCD has sides AB and DC that are equal and parallel. A student wrote the following sentences to demonstrate that quadrilateral ABCD is a parallelogram: Since side AB is parallel to side DC, the alternate interior angles, angle ABD and angle CDB, are congruent. Side AB is equal to side DC, and side DB is a common side to triangles ABD and BCD. Thus, triangles ABD and BCD are congruent by the SAS postulate. By CPCTC, angles DBC and ADB are congruent, and sides AD and BC are also congruent. Angles DBC and ADB _______________. Therefore, AD is parallel and equal to BC. Hence, quadrilateral ABCD is a parallelogram because its opposite sides are equal and parallel. Which phrase best completes the student’s proof?

Answer

To demonstrate that quadrilateral ABCD is a parallelogram, the phrase that accurately completes the proof is “Angle DBC and angle ADB form a pair of alternate interior angles which are congruent.” Explanation: For a quadrilateral to be a parallelogram, its opposite sides must be equal and parallel. We already know that AB is parallel to DC and that AB equals DC. Now, we need to establish that AD is parallel to BC and that AD equals BC. The congruency established shows AD equals BC. Additionally, to prove that AD is parallel to BC, we need to show that the alternate interior angles are equal. Thus, the phrase ‘Angle DBC and angle ADB form a pair of alternate interior angles which are congruent’ is the one that completes the proof.

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