There isn't a verse in the Bible that explicitly says one-third of the angels fell. But generally, this view is based on a passage in the Book of Revelation dealing with the great red dragon that many Christian commentators believe references the devil.
“And his tail swept away a third of the stars of heaven, and threw them to the earth. And the dragon stood before the woman who was about to give birth, so that when she gave birth he might devour her child,” (Rev. 12:4).
Stars are often used in Jewish literature symbolically to refer to angels since both the stars and the angels can in different senses be called "the hosts of heaven." The Book of Revelation seems to possibly use this kind of imagery in other passages, for example, when writing to the seven churches at the beginning of the book, seven stars are seen in Jesus' hand. It then explains:
"The seven stars are the angels of the seven churches," (Revelation 1:20b).
Or when the fifth trumpet is blown, we are told that:
"I saw a star from heaven which had fallen to the earth; and the key of the bottomless pit was given to him," (Revelation 9:1b).
This fallen "star" is described in personal terms and may also refer to an angel.
The Book of Revelation also refers to stars in other passages in ways that cannot be understood as angels, however, and so Christian commentators interpret the "one-third of the stars" in Revelation 12:4 in a variety of ways. The Bible, therefore, may or may not support the idea that one-third of the angels fell with Satan, depending on precisely what John meant in Revelation 12:4.