Does a divorce pronounced in anger take effect if the husband intended it in his heart and verbalized it to his mother, but not to his wife, and does this necessitate a tahlil marriage if they wish to reconcile?

If you utter a divorce while being aware of what you are saying, then your divorce is effective. However, a mere intention without utterance does not constitute a divorce.

Regarding informing your mother about divorcing your wife: - If you intended it to initiate a third divorce, and your wife is still under your legal custody, then it is a third divorce that constitutes an irrevocable major divorce (baynunah kubra). - If it was an announcement about a previous divorce, then no further divorce occurs by it.

Based on the latter possibility, the marriage is still valid if you took her back after the second divorce. If the waiting period (iddah) expired before you took her back, then you do not have the right to take her back except with a new contract.

Divorce is not a trivial matter, and one should not resort to it unless all means of reconciliation have been exhausted. Reconciling between spouses and treating each other kindly is preferable to separation, especially if they have children.