ABCD is a quad. in which AB = CD and AD = BC ? Show that it is a //gm .

Question

ABCD is a quad. in which AB = CD  and AD = BC ? Show that it is a //gm .

Jayden Jackson · Accepted Answer

Please see below.

In quadrilateral #ABCD#, we have #AB=CD# and #AD=BC#.

Now join segment #AC#.

Now in #DeltaABC# and #DeltaACD#, we have

#AB=CD# ---------------- already given

#AD=BC# ---------------- already given

and #AC=AC# ---------------- common

Hence #DeltaABC-=DeltaACD#

#:./_DAC=/_BCA# - both opposite equal sides #DC# and #AB#, and

#:.CD#||#AB# - as alternate angles are equal

#:./_DCA=/_CAB# - both opposite equal sides #AD# and #BC#

#:.AD#||#BC# - as alternate angles are equal

As opposite sides of quadrilateral are parallel,

#ABCD# is a parallelogram.

#Q.E.D.#

Parker Allen · Answer

undefined Given that ABCD is a quadrilateral in which AB = CD and AD = BC, we need to show that it is a parallelogram.

To prove that ABCD is a parallelogram, we can use the properties of opposite sides of a parallelogram:

1. Since AB = CD and AD = BC, opposite sides are equal.
2. Opposite sides of a parallelogram are parallel.

Therefore, by showing that the opposite sides of ABCD are equal, we can conclude that ABCD is a parallelogram.

We can represent the quadrilateral ABCD as follows:
ABCD

Given: AB = CD and AD = BC

To prove: ABCD is a parallelogram

Proof:
1. AB = CD (Given)
2. AD = BC (Given)
3. By definition, a parallelogram has opposite sides equal.
4. Therefore, AB || CD and AD || BC (Properties of parallelograms)
5. By definition, a parallelogram has opposite sides parallel.
6. Thus, ABCD is a parallelogram by definition.