In the given figure, AB is a chord of a circle with centre O and AB is produced to C such that BC = OB. Also, CO is joined and produced to meet the circle in D. If ∠AOC = 25° ∠ACD = 25°, then ∠AOD = ?

Question

Philoid · Accepted Answer

Given: BC = OB and ∠ACD = 25°

Here in Δ OBC

∠BOC = ∠BCO (angles opposite to equal sides are equal)

∴ ∠BOC = 25°

By angle sum property

∠BOC + ∠BCO + ∠OBC = 180°

25° + 25° + ∠OBC = 180°

50° + ∠OBC = 180°

∠OBC = 180° – 50°

∴ ∠OBC = 130°

Here

∠ABC = ∠ABO + ∠OBC = 180°

∠ABO + 130° = 180°

∠ABO = 180° – 130°

∴ ∠ABO = 50°

Now, in ΔAOB

OB = OA (radius)

∠ABO = ∠BAO = 50° (angles opposite to equal sides are equal)

By angle sum property

∠ABO + ∠BAO + ∠AOB = 180°

50° + 50° + ∠AOB = 180°

∠AOB = 180° – (50° + 50°) = 180° – 100° = 80°

∴ ∠AOB = 80°

Here

∠DOC = ∠AOD + ∠AOB + ∠BOC = 180°

∠AOD + 80° + 25° = 180°

∠AOD + 105° = 180°

∠AOD = 180° – 105°

∠AOD = 75°

∴ ∠AOD = 75°