How are Vedas related to the BhagavadGita?

Vedas are related to Gita in such a way that you can find all the teachings of Veda in Gita. Even the Supreme Himself has explained about this in Gita, "And I am seated in the heart of all beings. I am the source of memory, knowledge, and forgetfulness. I alone am to be known by the Vedas. I am indeed the author of the Vedanta as well as the knower of the Vedas" (15:15) You can find the teachings of Gita reflected in all the great scriptures of the world. The Bhagavad-gītā comprises chapters 25–42 of ‘Bhīṣma Parva’, the sixth section of the Mahābhārata. It appears as a part in the great epic Mahabaratha, yet it teaches us the lessons of life and also about moksha (liberation). After revealing the true knowledge to Arjuna, the Supreme gives the choice to Arjuna and asks him to proceed further in line with his decisions. It (freedom of choice) shows you the right path but gives you the control to decide. It reinforces that you are the master of your fate and the captain of your soul. "Thus, I have revealed to you the knowledge, the mystery of the mysteries. Reflect on it fully, and do as you wish." (18.63) It's not easy to answer how Gita  is related to vedas in one quora question.  But if you want to truly taste the nectar of entire Gita, I would suggest you to get this wonderful vedic book online, Bhagavad Gita - The Transcendental Knowledge.
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Bhagavad Gita

Bhagavad Gita, the "Song of the Supreme," was revealed by the great sage Kṛṣṇa Dvaipāyana or Veda Vyasa. The 700 verses in it consist of prominent truths from all great scriptures of the world. It is without a doubt that one who reads it with the utmost devotion finds ways to attain perfection and liberation. The world is torn by conflicts and miseries presently. The life of an individual in today's world is packed with problems, chaos, restlessness, and unhappiness. And every human being is looking for ways to end the everlasting misery. All beings are truly the forms of the divine. But due to illusion and ignorance, we are trapped in cyclic births and rebirths. We need Gita to bring back the divine connection that exists in us. The transcendental knowledge in Gita helps human beings to escape from this state of misery and to enter the union of the Supreme. Moreover, in this modern world, we need to understand that the battle for us is not on the battlefield, but it is in us. It's not about victory or defeat but understanding one's own "Self" in the whole process. Reading Bhagavad Gita is thus essential to make our self-discovery easier and simpler. With a lot of commentaries and interpretations for Gita, one may worry about finding the original Bhagavad Gita. The revealer of Bhagavad Gita is the great sage Veda Vyasa and only by reading the original version, one can taste the nectar and essence of Gita. The original Bhagavad Gita with authentic Sanskrit verses and exact translations in Hindi and English are found in ‘The Bhagavad Gita - The Transcendental Knowledge.’ The verses are carefully compiled and presented from a non-sectarian standpoint, taking into account the Sanskrit commentaries from the rich variety of traditions in Sanatana Dharma. The book comes in an antique book-styled box, adorned with Swarovski crystals and an attractive metal lock.