Question

Prove that the line segment joining the end point of two congruent arcs of a circle are either equal or parallel.​

Answer 1

Construction: Through O draw OR || BA or OR || CD as AB and CD are parallel tangents.
Proof:
OPA = 90o (radius is always perpendicular to tangent)
Since OR || BA (By construction)
OPA + POR = 180o
POR = 180o - 90o = 90O
Similarly QOR = 90o
POR + QOR = 180o
PQ is straight line through O. So PQ is diameter.

Answer 2

to

prove: AOB is a straight line passing through O.

Let PAQ and RBS be two parallel tangents to a circle with centre O.

Join OA and OB. Draw OC ∣∣ PQ.

Now, PA ∣∣ CO

⇒ ∠PAO+∠COA=180o [Sum of the angle on the same side of a transversal is 180o]

⇒ 90o+∠COA=180o [∵∠PAQ=angle between a tangent and radius=90o]

⇒ ∠COA=90o

Similarly, ∠COB=90o

∴∠COA+∠COB=90o+90o=180o