Question

In isosceles triangle ∆ABC,

Answer 1

Check the picture below.

the triangle is an isosceles, meaning it has two twin sides, that make two twin angles, namely AB = BC, and the angles at A and C are twins as well.

now, the AD segment, is an angle bisector, meaning it cuts the angle at A in two equal halves, so if say the angles are x° each, then the bisectors cuts it in (x/2)°.

now, we know ∡ADB is 110°, therefore its supplementary angle, in green, is 70°.

recall that the sum of all interior angles in a triangle is 180°, thus

$\bf 70~+~\cfrac{x}{2}+x=180\impliedby \textit{let's multiply both sides by the LCD 2} \\\\\\ 2\left( 70~+~\cfrac{x}{2}+x \right)=2(180)\implies 140+x+2x=360 \\\\\\ 3x=220\implies x=\cfrac{220}{3}\implies x=73\frac{1}{3}\\\\ -------------------------------\\\\ \measuredangle B+x+x=180\implies \measuredangle B+\cfrac{220}{3}+\cfrac{220}{3}=180 \\\\\\ \measuredangle B=180-\cfrac{440}{3} \implies \measuredangle B=\cfrac{540-440}{3}\implies \measuredangle B=\cfrac{100}{3} \\\\\\ \measuredangle B=33\frac{1}{3}$