Question

Which criteria must be true to conclude that ABCD is a parallelogram?

AB¯¯¯¯¯≅DC¯¯¯¯¯ and AC¯¯¯¯¯≅BD¯¯¯¯¯

AB¯¯¯¯¯≅DC¯¯¯¯¯

AC¯¯¯¯¯≅BD¯¯¯¯¯

AB¯¯¯¯¯≅DC¯¯¯¯¯ and AD¯¯¯¯¯≅BC¯¯¯¯¯

Answer 1

AB¯¯¯¯¯≅DC¯¯¯¯¯ and AD¯¯¯¯¯≅BC¯¯¯¯¯

Step-by-step explanation:

We know that the opposite sides of a parallelogram must be parrallel (and therefore the same length), so concluding that ABCD is a parallelogram based on AB¯¯¯¯¯≅DC¯¯¯¯¯ or AC¯¯¯¯¯≅BD¯¯¯¯¯ alone is not enough. That leaves us with two options left

1) AB¯¯¯¯¯≅DC¯¯¯¯¯ and AC¯¯¯¯¯≅BD¯¯¯¯¯

2) AB¯¯¯¯¯≅DC¯¯¯¯¯ and AD¯¯¯¯¯≅BC¯¯¯¯¯

Additionally, we can cancel out the first option as well. The AC¯¯¯¯¯≅BD¯¯¯¯¯ portion of the statement implies that the parallelogram would be a rectangle. And while we know that rectangles are parallelograms, not all parallelograms are rectangles. Option 1 is too restricting, and therefore our answer is AB¯¯¯¯¯≅DC¯¯¯¯¯ and AD¯¯¯¯¯≅BC¯¯¯¯¯.

Answer 2

AB ≅ DC and AD ≅ BC

Step-by-step explanation:

Given :

AB and DC are two opposite sides of the quadrilateral.

AD and BC are two opposite sides of the quadrilateral.

we have learnt, that a quadrilateral is a parallelogram if the opposite pairs of sides are congruent .

Then

The criteria AB ≅ DC and AD ≅ BC ,allow us to conclude that ABCD is a parallelogram.