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# 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° and ∠Bay and ∠Apy - ICSE Class 10 - Mathematics

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ConceptTangent Properties - If a Line Touches a Circle and from the Point of Contact, a Chord is Drawn, the Angles Between the Tangent and the Chord Are Respectively Equal to the Angles in the Corresponding Alternate Segments

#### Question

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° and ∠BAY and ∠APY

#### Solution

In ΔAXB,
∠XAB + ∠AXB + ∠ABX=180° [Triangle property]
⇒  ∠XAB + 50° + 70° = 180°
⇒  ∠XAB = 180° − 120° = 60°
⇒  ∠XAY=90° [Angle of semi-circle]
∴ ∠BAY = ∠XAY − ∠XAB = 90° − 60° = 30°
and  ∠BXY =  ∠BAY = 30° [Angle of same segment]
∠ACX =  ∠ BXY +  ∠ABX [External angle = Sum of two interior angles]
= 30° + 70°
= 100°
also,
∠XYP = 90° [Diameter ⊥ tangent]
∠ APY = ∠ACX − ∠CYP
∠APY = 100° − 90°
∠APY = 10°

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Solution 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° and ∠Bay and ∠Apy Concept: Tangent Properties - If a Line Touches a Circle and from the Point of Contact, a Chord is Drawn, the Angles Between the Tangent and the Chord Are Respectively Equal to the Angles in the Corresponding Alternate Segments.
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