AB = 3 + x

DC = 4x

AD = y + 1

BC = 2y

Quadrilateral ABCD is a parallelogram if both pairs of opposite sides are congruent. Show that quadrilateral ABCD is a parallelogram by finding the lengths of the opposite side pairs.

The given quadrilateral ABCD is a parallelogram since the opposite sides are of same length AB and DC is 4 and AD and BC is 2.

Step-by-step explanation:

ABCD is a quadrilateral with their opposite sides are congruent (equal).

The both pairs of opposite sides are given as AB = 3 + x, DC = 4x, AD = y + 1, BC = 2y.

AB and DC are opposite sides and have same measure of length. AD and BC are opposite sides and have same measure of length.

To find the length of AB and DC:

AB = DC

3 + x = 4x

Keep x terms on one side and constant on other side.

3 = 4x - x

3 = 3x

x = 1

Substiute x=1 in AB and DC,

AB = 3+1 = 4

DC = 4 (1) = 4

To find the length of AD and BC:

AD = BC

y + 1 = 2y

Keep y terms on one side and constant on other side.

2y-y = 1

y = 1

Substiute y=1 in AD and BC,

AD = 1+1 = 2

BC = 2 (1) = 2

Therefore, the opposite sides are of same length AB and DC is 4 and AD and BC is 2. The given quadrilateral ABCD is a parallelogram.