Question

an isosceles triangle is inscribe in aradius 100 in .find the angles of the triangle if it is base subtends an arc of 143.7 in

Answer 1

I will assume you meant to say
" ... is inscribed in a circle with radius 100 inches ..."

Make a sketch, draw radii from the base to the centre of the circle.
We will use the fact that the central angle subtended by an arc is twice the subtended angle at the circle

For the central angle:

arc = r Ø , where Ø is in radians
so 143.7 = 100Ø
Ø = 1.437 radians

then the angle on the circle = .7185 radians
and the two equal base angles must be (π - .7185)/2 = 1.212 radians

btw, in degrees that would be 41.167° , 69.416° , 69.416°