In the given quadrilateral ABCD, ∠DAB = ∠CBA, OA = 3 cm and AD = BC. What is the length of OB?

Consider ΔBAD and ΔABC,

AD = BC   [Given]

∠DAB = ∠CBA   [Given]

AB = AB   [Common]

So, ΔBAD ≅ Δ ABC   [By SAS congruency criterion]

∴ ∠ABD = ∠BAC   [By CPCT]

⇒ ∠ABO = ∠BAO

Thus, in ΔOAB, we have ∠ABO = ∠BAO

It is known that the sides opposite to the equal angles are equal.

Therefore, OA = OB = 3 cm