Question

In a given figure, AE = DE and BC || AD. Prove that the points A, B, C and D are concyclic. Also, prove that the diagonals of the quadrilateral ABCD are equal.

Answer 1

Where is the figure please?

Answer 2

Intended Answer:

ABCD is a cyclic quadrilateral.
The diagonals mentioned are equal.
Figure is given below.

Given:- AE = DE and BC parallel to AD.

Prove:-

1. ABCD is a cyclic quadrilateral.
2. The diagonals of ABCD are equal.

Step-by-step explanation:

By given, Angle BAD = Angle CDA = x (Angles opposite to equal sides are equal.)

=> ABCD is isosceles trapezium.

Angle A + Angle B = 180 = Angle C + Angle D

=> Angle B = 180 - x

=> Angle C = 180 - x

W.K.T - If opposite angles sum up to 180 degrees, it is a cyclic quadrilateral.

=> Angles (A + C) = Angles (B + D) = 180 degrees

∴ ABCD is a cyclic quadrilateral. Hence proved (i).

In ΔBAD and ΔCDA,

Angle  A = Angle D (Given)
AB = CD (Given)
AD (Common)

ΔBAD ≅ ΔCDA (SAS rule)

∴ By CPCT, BD = AC, both lines are diagonals. Hence proved (ii).

∵ We have proved the required.

This answer is written from the perspective of Grade 9 CBSE, adapted to the 2021 - 2022 syllabus.