Please keep drawing the figure as you follow below.

Let ABC be the triangle with sides BC, AC, AB opposite angles A, B, C being in length equal to a, b, c respectively.

Draw perpendiculars AD, BE and CF from A, B, C to opposite sides meeting sides BC, AC and AB at D, E, F respectively.

Now perpendicular AD² = AC² - CD² ==> AD² < AC² or AD < AC

or AD < b -----(1)

Also, BE² = AB² - AE² ==> BE² < AB² or BE < AB

or BE < c ------(2)

Likewise , CF² = BC² - BF² ==> CF² < BC² or CF < BC

or CF < a ----(3)

Adding inequalities (1), (2), (3), AD + BE + CF < a + b + c