One who missed part of the prayer does what the imam does, and when the imam gives the taslim he stands and completes the rest of his prayer

Sayyiduna Abu Hurayrah (may Allah be pleased with him) said that the Messenger of Allah (peace and blessings be upon him) said: "When the call is made for the prayer, come to it walking with calm and tranquillity; whatever you catch, pray it, and whatever you have missed, complete it."

(a) Urwah bin al-Mughirah bin Shu'bah (may Allah have mercy on him) relates that al-Mughirah bin Shu'bah (may Allah be pleased with him) went on a military expedition with the Messenger of Allah (peace and blessings be upon him). Then he mentioned the complete hadith about the ablution of the Messenger of Allah (peace and blessings be upon him) and his wiping over the leather socks. He said: Then he came back, and Sayyiduna al-Mughirah (may Allah be pleased with him) said: I came with him until he (peace and blessings be upon him) found that the people had put Sayyiduna Abd al-Rahman bin Awf (may Allah be pleased with him) forward to lead the prayer. Abd al-Rahman bin Awf (may Allah be pleased with him) led them in prayer, and the Messenger of Allah (peace and blessings be upon him) caught only one rak'ah. He prayed the second rak'ah with the people, and when Sayyiduna Abd al-Rahman bin Awf (may Allah be pleased with him) said the salam, he (peace and blessings be upon him) stood up to complete his prayer. The Muslims were alarmed when they saw this, and they began saying "Subhan Allah" (Allah is free of all imperfection) a great deal. When the Messenger of Allah (peace and blessings be upon him) had completed his prayer, he turned towards them and said: "You did well," or he said: "You have done what is right." He (peace and blessings be upon him) was pleased with them for having offered the prayer on time.
(b) In the narration of Ibn Jurayj (may Allah have mercy on him), Sayyiduna al-Mughirah (may Allah be pleased with him) said: I wanted to move Sayyiduna Abd al-Rahman bin Awf (may Allah be pleased with him) back, but the Prophet (peace and blessings be upon him) said: "Leave him."

Sayyiduna Mu'adh bin Jabal (may Allah be pleased with him) said: The prayer passed through three stages. Then he mentioned the stage of the qiblah and the stage of the adhan, and said: These are two stages. He said: People would come to the prayer when the Prophet (peace and blessings be upon him) had already got ahead of them by part of the prayer, so someone would indicate to them with his fingers how much he had prayed: one, two. Then Sayyiduna Mu'adh (may Allah be pleased with him) came when the Prophet (peace and blessings be upon him) had already got ahead of him by part of the prayer, and he said: In whatever state I find him, I will join him in it and then make up (what I missed). So he entered the prayer, and when the Messenger of Allah (peace and blessings be upon him) finished, Sayyiduna Mu'adh (may Allah be pleased with him) stood up to make up (what he had missed). The Messenger of Allah (peace and blessings be upon him) said: "Mu'adh has set a practice for you, so do likewise."
This chain is broken, because Abd al-Rahman bin Abi Layla did not meet Mu'adh; the version of Shu'bah, in which Abd al-Rahman says "Our companions narrated to us", is more correct.

It is narrated from a man of the Ansar (may Allah be pleased with him) that a man came while the Prophet (peace and blessings be upon him) was praying, and he (peace and blessings be upon him) heard the sound of his footsteps. When he (peace and blessings be upon him) had finished the prayer with the salam, he said: "Which of you entered (the prayer)?" The man said: I did, O Messenger of Allah. He (peace and blessings be upon him) asked: "In what state did you find us?" He said: In prostration. He (peace and blessings be upon him) said: "Do likewise: when you find the imam standing, bowing, prostrating or sitting, do as you find him doing; and if you did not catch the bowing, do not count that rak'ah."

(a) Nafi' (may Allah have mercy on him) said: When Sayyiduna Ibn Umar (may Allah be pleased with them both) found that the imam had already prayed part of the prayer, he would pray with the imam whatever he caught. If the imam stood, he stood, and if the imam sat, he sat, until the imam completed his prayer. He would not differ from the imam but would follow him.
(b) Nafi' (may Allah have mercy on him) said that Sayyiduna Ibn Umar (may Allah be pleased with them both) used to say: When you miss the bowing, you have missed the rak'ah.

Sayyiduna Ibn Umar (may Allah be pleased with them both) said: "In whatever state you find the imam, be in that state."

(a) Nafi' (may Allah have mercy on him) said that when Sayyiduna Abdullah bin Umar (may Allah be pleased with them both) missed a rak'ah or the like, he would stand up as soon as (the imam) said the salam, and he did not wait for the imam to stand.
(b) Sayyiduna Abu Sa'id al-Khudri (may Allah be pleased with him) said: This is the Sunnah.