Question

The bisector of an angle of a triangle divides the opposite sides into segments that are proportional to the adjacent sides.

• A. True
• B. False

Answer 1

The correct option is A

True

If we draw BE parallel to DA,meeting CA extended to E;then

In $△CAD$ and $△CEB$

$\angle BCA=\angle BCE$ {same angle}

$\angle EBC=\angle ADC$ {because AD is parallel EB by construcyion and they are cut by transversal AB}

$\angle BEC=\angle DAC$ {because AD is parallel EB by construcyion and they are cut by transversal CE}

Now,we can say

$△CAD\cong △CEB$ {AAA similarity}

$\frac{CE}{AC}=\frac{BC}{DC}$

$\frac{AE+AC}{AC}=\frac{BD+DC}{DC}$ {AE=AB by construction}

$\frac{AB+AC}{AC}=\frac{BD+DC}{DC}$

$\frac{AB}{AC}+1=\frac{BD}{DC}+1$

$\frac{AB}{AC}=\frac{BD}{DC}$

So,the bisector of an angle of a triangle divides the opposite sides

into segements that are proportional to the adjacent sides.