Question

3. In cyclic quadrilateral, diagonals
intersect each other at right
angles and one of the pairs of
consecutive side is congruent
show that the remaining pair of
consecutive sides is congruent

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Answer 1

Proved below.

Step-by-step explanation:

Given:

We are given that two consecutive angles of cyclic quadrilateral are congruent.

As shown in tha figure below,

Let ∠ABC = ∠BCD

∠BAD + ∠BCD = 180°  (ABCD is a cyclic quadrilateral)

∠BAD + ∠ABC=  180°    (∠ABC = ∠BCD)

The sum of angles on the same side of a transversal line is supplementary.

∴ AD║BC

Construction:

Draw a line from D parallel to AB which intersects BC at E.

AD║BC and  AB║DE

∴ ABED is a parallelogram.

So, AB = DE       [1]            (Opposite sides of a parallelogram are equal)

∠ABE = ∠DEC   (Corresponding angles)

∠ABC = ∠BCD  (Given)

So, ∠DEC = ∠BCD

∴ DE = DC            [2]           (Sides opposite to equal angles are equal)

Using equations (1) and (2), we get

AB = DC

Therefore, the remaining pair of  consecutive sides is congruent.

Hence proved.