Question

In the given figure O is the centre of the circle. If diameter AD bisects ∠BOC. Prove that ∠BOD and ∠OBC are complementary angles. Explain with full steps.

Points : 10

Answer 1

In triangle OBD and ODC
angle BOD = angle DOC ( given)
BO = OC ( radius)
OX = OX ( common)
So triangle OBD congruent to ODC
so BX = XC ( CPCT )
And since OD IS Bisecting BC angle OXB =90 and so angle OBD + angle DOB = 90 ( angle sum property of triangle)

Answer 2

AD diameter bisects the chord perpendicularly.take the intersecting point P.
therefore
triangle BPO is right angled triangle
<BPO=90
<OBP+<BOP++<BPO=180
<OBP+<BOD+90=180
<OBC+<BOD=90
sum of any two angles equals 90 degrees are called complementary