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# In the Given Figure, Two Circles Touch Each Other Externally at Point P. Ab is the Direct Common Tangent of These Circles. Prove That: (Ii) Angles Apb = 90° - ICSE Class 10 - Mathematics

ConceptTangent Properties - If Two Circles Touch, the Point of Contact Lies on the Straight Line Joining Their Centers

#### Question

In the given figure, two circles touch each other externally at point P. AB is the direct common tangent of these circles. Prove that:

(ii) angles APB = 90°

#### Solution

ii) Now in Δ ATP ,
∴ ∠TAP = ∠TPA
Similarly in Δ BTP,∠TBP = ∠TPB
∠TAP +∠TBP =∠APB
∴ TAP + ∠TBP  + ∠APB =180°
⇒ ∠APB =  ∠TAP  + ∠TBP =90°