Question

Please answer correctly......
in the following figure ab is the common tangent to circle C1 and C2 C1 and C2 are touching externally at c a d and DC are two chords of a circle C1 and B and C are two chords of a circle to find the measure of angle ADC + angle CEB​

Answer 1

Given :  ab is the common tangent to circle C1 and C2

C1 and C2 are touching externally at c

AD and DC are two chords of a circle C1 and

BE and CE are two chords of a circle

To find  :  the measure of angle ADC + angle CEB​

60°

90°

120°

can not be determined

Solution:

Let say X and Y are the centers of the circle

Join XA , YB , XY

XY  will pa ss through C   as circles are touching at C hence having common tangent at point C

so we get

ABYX  a quadrilateral

∠A = ∠B = 90°   ( ∠XAB = ∠YBA  = 90°  tangents )

∠A + ∠B   = 180°  

Hence  ∠AXY +  ∠BYX  = 180°  ( as sum of angles of quadrilateral is 360°)

C lies on XY

=> ∠AXC  +  ∠BYC  = 180°

∠AXC = 2 ∠ADC

∠BYC  = 2 ∠CEB

=> 2 ∠ADC  + 2 ∠CEB  = 180°

=> ∠ADC  +  ∠CEB  = 90°

the measure of angle ADC + angle CEB​  is 90°

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