A flexibly engaged continuously variable transmission gearbox, comprising the following components connected sequentially in the transmission gearbox: a power input shaft conical roller, a sectional input flexible sprocket comprising five sectional input flexible sprocket members, a tooth-shaped chain, a sectional output flexible sprocket comprising five sectional output flexible sprocket members, a power output shaft conical roller; when the power input shaft conical roller rotates, it drives the five sectional input flexible sprocket members that are slidably matched in five equidistant T-shaped grooves on the input shaft conical roller to rotate, then the sectional input flexible sprocket drives the tooth-shaped chain engaged therewith to transmit, and thus drives the five sectional output flexible sprocket members that are slidably matched in five equidistant T-shaped grooves on the output shaft conical roller to rotate, with the five sectional output flexible sprocket members connected slidably to the output shaft conical roller through the five equidistant T-shaped grooves, thus driving the output shaft conical roller to rotate to output power, characterized in that the variable transmission is achieved by engagement of the tooth-shaped chain with the sectional flexible sprockets (2, 4), wherein the sectional flexible sprockets (2, 4)with an equal pitch of teeth, respectively composed of five sectional T-shaped sprocket members, are sliding in five equidistant T-shaped grooves on input and output shaft conical rollers (1, 5) such that the sectional flexible sprockets (2, 4), by sliding in different engagement diameter positions on the conical rollers (1, 5), can make the engagement linear velocity of the flexible sprockets (2, 4) and the tooth-shaped chain become greater or less, the larger the diameter the greater the linear velocity, the smaller the diameter the less the linear velocity, so that output shaft conical roller rotational speed is changed. a power input shaft conical roller, a sectional input flexible sprocket comprising five sectional input flexible sprocket members, a tooth-shaped chain, a sectional output flexible sprocket comprising five sectional output flexible sprocket members, a power output shaft conical roller; when the power input shaft conical roller rotates, it drives the five sectional input flexible sprocket members that are slidably matched in five equidistant T-shaped grooves on the input shaft conical roller to rotate, then the sectional input flexible sprocket drives the tooth-shaped chain engaged therewith to transmit, and thus drives the five sectional output flexible sprocket members that are slidably matched in five equidistant T-shaped grooves on the output shaft conical roller to rotate, with the five sectional output flexible sprocket members connected slidably to the output shaft conical roller through the five equidistant T-shaped grooves, thus driving the output shaft conical roller to rotate to output power, characterized in that the variable transmission is achieved by engagement of the tooth-shaped chain with the sectional flexible sprockets (2, 4), wherein the sectional flexible sprockets (2, 4)with an equal pitch of teeth, respectively composed of five sectional T-shaped sprocket members, are sliding in five equidistant T-shaped grooves on input and output shaft conical rollers (1, 5) such that the sectional flexible sprockets (2, 4), by sliding in different engagement diameter positions on the conical rollers (1, 5), can make the engagement linear velocity of the flexible sprockets (2, 4) and the tooth-shaped chain become greater or less, the larger the diameter the greater the linear velocity, the smaller the diameter the less the linear velocity, so that output shaft conical roller rotational speed is changed.