AB and CD are two diameters of a circle. Prove that ADBC is a rectangle

Explanation

AOB and COD are two diameters of the circle of the circle with centre at O.
To prove : ADBC is a recangle
Construction: Let us join `A.D, D, B, B, C and C,A`.
Proof : AOB is a diameter `:. angle ACB and angle ADB` are two semi-circular angles.
`:. angle ACB`= 1 right angle and `angleADB=1` right angle.
`:.` Two opposite angles of the quadrilateral ADBC are both right angles.
Similarly, COD is a diameter `:. angle CAD and angle CBD` are both semi-circular angles.
`:. angle CAD=1` right angle and `angleCBD=1` right angle.
`:.` Two other opposite angles of the quadrilateral ADBC are also right angles.
Again, in `Delta AOD and Delta BOC, OA= OB, OD= OC [ :.` radii of same circle] and `angle COD= angleBOC [ :.` opposite angles]
`:. Delta AOD~= Delta BOC` [by the S-A-S condition of congrency]
`:. AD=BC [ :.` similar sides of congruent triangles]
Similarly, it can be proved that `AC=BD`
`:.` opposite sides of the quadrilateral are equal and each of its angles is a right angle.
Hence ADBC is a rectangle.