Verified answer

[tex]\large\underline{\sf{Solution-}}[/tex]

Given that, ABCD is a parallelogram.

Further given that, P is the midpoint of AB.

[tex]\implies\bf\:AP = PB = \dfrac{1}{2} AB - - - (1) \\ [/tex]

Further given that, Q is the midpoint of CD.

[tex]\implies\bf\:CQ = QD = \dfrac{1}{2} CD - - - (2) \\ [/tex]

Now, As ABCD is a parallelogram. We know, Opposite sides are equal.

[tex]\implies\sf\:AB = CD \\ [/tex]

[tex]\implies\sf\:\dfrac{1}{2} AB = \dfrac{1}{2} CD \\ [/tex]

[tex]\implies\bf\:AP = QD - - - (3) \\ [/tex]

Again, ABCD is a parallelogram. Opposite sides are parallel.

[tex]\implies\sf\:AB \: \parallel \: CD \\ [/tex]

[tex]\implies\bf\:AP \: \parallel \: QD - - - (4) \\ [/tex]

From equation (3) and (4), we get

[tex]\implies\bf\:AP \: \parallel \: QD \: \: and \: \: AP = QD \\ [/tex]

We know, In a quadrilateral, if one pair of opposite sides are equal and parallel, then it is a parallelogram.

[tex]\implies\bf\:APQD \: is \: a \: parallelogram \\ [/tex]