Question

1.Tangents AP and AQ are drawn to circle with centre O, from an external point A . Prove that ∠PAQ = 2. ∠OPQ

2.A quadrilateral ABCD is drawn to circumscribe a circle. Prove that AB + CD = AD +BC.​

Answer 1

1.⤵

Solution:

∠POQ + ∠PAQ = 180˚ [Angle between tangents + angle in the centre = 180˚] …………(1)

In ∆POQ , ∠POQ + ∠OPQ + ∠OQP = 180˚ [Sum of the angles of a triangle = 180˚]

Therefore,

∠POQ + 2∠OPQ = 180˚ [ Since ∠OPQ = ∠OQP] ………(2)

From (1) and (2)

∠POQ + ∠PAQ = ∠POQ + 2∠OPQ

∠PAQ = 2∠OPQ

2.⤵

AP = AS = l

BS = BR = m

CR = CQ = n

DP = DQ = o

AB + CD = AS + SB + CQ + QD = l + m + n + o …………….(1)

AD + BC = AP + PD + BR + RC

l + m + n + o = l + o + m + n ……………(2)

From (1) and (2),

AB + CD = AD + BC

Answer 2

please see the attachment