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In the figure XY and X'Y' are two parallel tangents to a circle with centre O and another tangent AB with point of contact C interesting XY at A and X'Y' at B, what is the measure of ∠AOB

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Solution:

Join OC 

In Δ OPA and Δ OCA 

OP = OC (radii of same circle) 

PA = CA (length of two tangents from an external point) 

AO = AO (Common) 

Therefore, Δ OPA ≅ Δ OCA (By SSS congruency criterion) 

Hence, ∠ 1 = ∠ 2 (CPCT) 

Similarly ∠ 3 = ∠ 4 

∠PAB + ∠QBA =180°(co interior angles are supplementary as XY∥X’Y’) 

2∠2 + 2∠4 = 180° 

∠2 + ∠4 = 90°-------------------------(1) 

∠2 + ∠4 +∠AOB = 180° (Angle sum property) 

Using (1), we get, ∠AOB = 90°

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