The bisector of an angle of a triangle divides the opposite sides into segments that are proportional to the adjacent sides.
True
If we draw BE parallel to DA,meeting CA extended to E;then
In △CAD and △CEB
∠BCA=∠BCE {same angle}
∠EBC=∠ADC {because AD is parallel EB by construcyion and they are cut by transversal AB}
∠BEC=∠DAC {because AD is parallel EB by construcyion and they are cut by transversal CE}
Now,we can say
△CAD≅△CEB {AAA similarity}
CEAC=BCDC
AE+ACAC=BD+DCDC {AE=AB by construction}
AB+ACAC=BD+DCDC
ABAC+1=BDDC+1
ABAC=BDDC
So,the bisector of an angle of a triangle divides the opposite sides
into segements that are proportional to the adjacent sides.