Question

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.

Answer 1

Answer: SSS postulate

Explanation:

I guarantee that the answer is this.

Side AB is parallel to side DC, so the alternate interior angles, angle ABD and angle CDB, 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 (SSS Postulate). By CPCTC, angles DBC and BDA are congruent and sides AD and BC are congruent. Angle DBC and angle BDA 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.

Answer 2

The ΔABD and ΔCDB are congruent by SAS concurrency.

Explanation:

First please take a look with diagram in attachment.

In ΔABD and ΔCDB

AB=CD { Given in question}

`\angle ABD=\angle CDB`

BD=DB {Common in both triangle}

Therefor, ΔABD and ΔCDB are congruent by SAS

`\angle DBC=\angle ADB` by CPCT

AD=CB by CPCT

But
`\angle DBC` and
`\angle ADB` are pair of alternate interior angle.

Therefore, AD parallel to CB (AD||CB) and AD=CB

We are given AB parallel to CD (AB||CD) and AB=CD

It means quadrilateral ABCD would be parallelogram because opposite sides are equal and parallel.