Question

PLEASE HELP!!! I need the answer and the work to go along with it.

Answer 1

----------------------------------------------------------------------------------------------------
Check the picture attached.

the secants of the circle, BA and DC, intersect at point P, as shown.

i)
let m(P)=a

ii)
let m(CDA)=b, then the measure of ar CA is 2b, as "the measure of an inscribed angle is half the measure of the arc it intercepts"

iii)
m(DAB)= m(P)+m(CDA)= a+b, as DAB is an exterior angle of the triangle PAD.

iv)
then the measure of arc DB is 2m(DAB)=2(a+b)=2a+2b

v)
$m(P)= \frac{(2a+2b)-2b}{2}= \frac{m(DB)-m(CA)}{2}$

This is always true,
----------------------------------------------------------------------------------------------------

In our problem, there is a slight difference, we have 1 secant and one tangent, but it is just a special case and nothing changes.

As an exercise, we can repeat the steps of the general proof, by first joining U and T.
----------------------------------------------------------------------------------------------------

by the theorem we just proved,

m(R)=(m(TU)-m(SU))/2

30°=(100°-m(SU))/2

60°=100°-m(SU)

m(SU)=40°