In the given figure, XY is the diameter of the circle and PQ is a tangent to the circle at Y.

If ∠AXB = 50° and ∠ABX = 70°, find ∠BAY and ∠APY.

In △AXB,

⇒ ∠AXB + ∠XAB + ∠ABX = 180° [Angle sum property of triangle]

⇒ 50° + XAB + 70° = 180°

⇒ ∠XAB = 180° - 120° = 60°.

From figure,

∠XAY = 90° [Angle in a semi-circle is a right angle.]

∠BAY = ∠XAY - ∠XAB = 90° - 60 = 30°.

∠BXY = ∠BAY = 30° [Angles in same segment are equal]

We know that,

An exterior angle is equal to the sum of two opposite interior angles.

⇒ ∠ACX = ∠BXC + ∠CBX

⇒ ∠ACX = ∠BXY + ∠ABX [From figure, ∠BXC = ∠BXY and ∠CBX = ∠ABX]

⇒ ∠ACX = 30° + 70° = 100°.

We know that,

Diameter is perpendicular to tangent.

⇒ ∠XYP = 90°

An exterior angle in a triangle is equal to sum of two opposite interior angles.

⇒ ∠ACX = ∠APY + ∠CYP

⇒ ∠APY = ∠ACX - ∠CYP = 100° - 90° = 10°.

Hence, ∠APY = 10° and ∠BAY = 30°.