The proof method of (1)supernuaahc is very good. The second question is a bit complicated. Now I'll give you the proof of the second question for reference, hoping it will help you.

(2)

Extend the point D where BO intersects AC, so that DE is vertical to OA, DF is vertical to AB, and point E and point F are vertical to foot.

Angle c = 80? Angle OBA = 30 Angle OAB = 10, triangle ABC is isosceles. We can calculate the angles CDB = 80°, EOA = 40° and OAD = 40°, which proves that the triangle BCD and the triangle ADO are isosceles, so BC=BD=5 and AE=OE= 1/2OA.

Is it easy to prove triangle ADE? Congruent triangles DAF? Because they are all right-angled triangles, one acute angle is 40 and the other acute angle is 50, and AD=AD. So we get DF=AE.

The middle angle DBF of the right triangle BDF is 30, and DF= 1/2BD can be obtained.

Because of AE? = 1/2OADF= 1/2BD？ DF=AE, so 1/2OA? = 1/2BD？ Get OA=BD=5.