in the given figure, arc AB congruent to arc CD. prove that angle A = angle B.

Dear student,

Following is the answer:
given that arc AB = arc CD

Therefore, $\angle AOB = \angle COD [congruent arcs of circles make equal angles at the center]$
$$\begin{gathered} \Rightarrow \angle AOB+\angle BOC =\angle COD + BOC \\ \Rightarrow \angle AOC = \angle BOD \end{gathered}$$
In triangle AOC and triangle BOD,
OA = OB
​OC = OD
$\angle AOC = BOD$
Hence, $triangle AOC \cong triangle BOD (by SAS criteria)$
Therefore, by CPCT we have $\angle A = \angle B (by CPCT)$