Question

A student wrote the following sentences to prove that quadrilateral ABCD is a parallelogram: Side AB is parallel to side DC so the alternate interior angles, angle ABD and angle BDC, are congruent. Side AB is equal to side DC and DB is the side common to triangles ABD and BCD. Therefore, the triangles ABD and CDB are congruent by _______________. By CPCTC, angles DBC and ADB are congruent and sides AD and BC are congruent. Angle DBC and angle ADB form a pair of alternate interior angles. Therefore, AD is congruent and parallel to BC. Quadrilateral ABCD is a parallelogram because its opposite sides are equal and parallel. Which phrase best completes the student's proof?

Answer 1

Answer:

SAS rule

Step-by-step explanation:

To prove: A quadrilateral ABCD is a parallelogram.

Proof: It is given that ABCD is an equilateral, thus side AB is parallel to side DC so the alternate interior angles are congruent, thus ∠ABD and ∠BDC, are congruent.

Also, From ΔADB and ΔCDB, we have

AB=DC (Given)

∠ABD =∠BDC (Alternate angles)

DB=DB (Common)

Thus, by SAS rule of congruency, ΔADB is congruent to ΔCDB that is ΔADB≅ΔCDB.

By CPCTC, ∠DBC and ∠ADB are congruent and sides AD and BC are congruent. ∠DBC and ∠ADB form a pair of alternate interior angles.

Therefore, AD is congruent and parallel to BC.

Quadrilateral ABCD is a parallelogram because its opposite sides are equal and parallel.

Hence proved.

Answer 2

Therefore, the triangles ABD and CDB are congruent by SAS