Question

Prove that the sum of the angles in the four segments exterior to a cyclic quadrilateral is equal to six right angles

Answer 1

let Abcd be the cyclic quadrilateral
angle p, angle q,r and s are angles in the four external segments.
to prove that angle p + angle q + angle r + angle s = six right angles
construction:
join sb and sc.
from the figure APBS is the cyclic quadrilateral
angle asb+ angle p = 180°.........(1)
similarly BQCS is cyclic quadrilateral
angle CSB+angle q =180°...........(2)
CRDS is cyclic quadrilateral
angle csd+angle r =180°..............(3)
now adding 1,2and 3 we get,
ANGLE ASB+ANGLE P + ANGLE CSB+ANGLE Q+ANGLE CSD + ANGLE R= 180+180+180.
angle p + angle q + angle r + (angle asb+ angle CSB + angle +angle CSB )=3×180
but angle asb + angle CSB + angle csb = angle s
angle p + angle q + angle r + angle s = 3×180
angle p + angle q + angle r + angle s=3×2×90
angle p + angle q + angle r + angle s=6×90
angle p + angle q + angle r + angle s= 6×right angles.