Geometry, Circles & Chords

Question

For circle O,  and m∠ABC = 55°.

In the figure, ∠AOB, ABO, or BOA and ∠BCO, OBC, or BOC have measures equal to 35°.

please select one of the underlined from each

Answer 1

mCD = 125, so since CA is diameter of circle (which divides it into two 180 arcs, mAD = 180 - 125 = 55.

<BOA is central angle which is equal to measure of intercepted arc AD, so m<BOA = 55.

Since AB is tangent to circle it meets at a right angle, so m<BAO = 90 and BAO is a right triangle whose angle measures sum to 180. So m<ABO = 180 - m<BOA - m<BAO = 180 - 55 - 90 = 35.

Since m<ABC = 55 and m<BAO = 90, then ABC is a right triangle whose angle measures sum to 180, m<BCO = 35.