Question

in the figure two circles with Centre O and P intersect each other at point B and C line a b intersect the circle with Centre O at point A and B and touches the circle with Centre p at point P prove angle A D E + Angle B C is equal to 180​.
plz answer

Answer 1

$\huge\blue{Answer}$

ABCD is a cyclic quadrilateral where ∠ADE + ∠BCE = 180°

Construction - Join CD.

∠CEB =∠CDE ....(1) [As angles are in the alternate segment]

Since, ABCD is a cyclic quadrilateral, then

∠ADC + ∠ABC = 180° (The opposite angles of cyclic quadilateral are supplementary) --- eq 2

∠CBE + ∠ABC = 180° (Linear pair) --- eq 3

From equation (2) and (3), -

∠ADC = ∠CBE --- eq 4

In ΔCBE,

∠CBE+∠CEB+∠BCE = 180° [Angle sum property]

=∠ADC + ∠CDE + ∠BCE = 180° [Using 1 and 4 ]

= ∠ADC + ∠CDE]+ ∠BCE = 180°

= ∠ADE + ∠BCE = 180°

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