Question

Draw a circle
a centre O, two diameters A B and C D, two radii OP and OQ , two chords PQ , XY, one Sector , one segment

Answer 1

Step-by-step explanation:

Draw seg PY,

seg PQ∥ seg XY and seg PY is their transversal.

∴∠XYP=∠YPQ  ....(alternate angles) ...(1)

Inscribed angle, ∠YPQ intercepts arc YQ.

∴∠YPQ=  

2

1

​  

m(arcYQ)     ....(by inscribed angle theorem)   ...(2)

Similarly, ∠XYP=  

2

1

​  

m(arcXP)    ....(3)

∴  

2

1

​  

m(arcXP)=  

2

1

​  

m(arcYQ)    ....[from (1), (2) and (3)]

∴m(arcXP)=m(arcYQ)

Now, m(arcXP)=∠XOP

m(arcYQ)=∠YOQ

∴∠XOP=∠YOQ     ....(4)

∴∠XOP=∠YOQ

i.e. ∠POX=∠QOY