Arriving Late to the Prayer Service
If one arrives to pray Mincha and finds that the congregation has already begun praying the Shemoneh Esrei he should not begin praying his own silent prayer unless he is able to finish before the prayer leader reaches kedusha in the repetition.
Since the entire congregation bows at the beginning and end of the modim blessing, even one who is not praying must bow with them. Accordingly, one who arrives after kedusha should not start praying unless he is able to finish before the prayer leader reaches the blessing of modim, so that he can bow with the rest of the congregation. One is also allowed to pray if he will reach modim or one of the other blessings where one bows (e.g. birkat Avot) together with the prayer leader, since in this way he will be able to bow together with everyone else.
The Rema writes that the rule of waiting applies to saying “Amen” to both the blessings of “Ha-kel Hakadosh” (the Holy
The above rules apply only when there will be enough time to pray afterwards. However, if it nearly sunset and there is not enough time to wait, one should pray right away. And if he happens to be in the middle of one of the blessings when the prayer leader says modim he should bow with everyone else. However, one does not bow if he is at the beginning or end of a blessing (except for modim or Avot).
During the Shacharit morning prayer, when one must connect the “Blessing of Redemption” with the Shemoneh Esrei, one does not wait to say kedusha, but instead begins the Shemoneh Esrei immediately after the Blessing of Redemption.
One may also wait before completing the blessing in order to answer kedusha, and afterwards complete the blessing, thereby fulfilling both the requirement of connecting Redemption with the Shemoneh Esrei as well as answering kedusha. (Shulchan Aruch HaRav)
During the Ma’ariv prayer one who comes late must be sure he can finish the Shemoneh Esrei before the kaddish that is recited immediately afterwards by the prayer leader. If not, he should not pray until after answering kaddish. (Shulchan Aruch Orach Chaim 109:1)