NCERT Solutions for Class 9 Maths Chapter 10
Important NCERT Questions
Circles
NCERT Books for Session 2022-2023
CBSE Board, UP Board and Other state Boards
EXERCISE 10.5
Page No:185
Questions No:8
If the non-parallel sides of a trapezium are equal, prove that it is cyclic.
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Given: In Trapezium ABCD, AB ∥ DC and AD = BC.
Construction: Draw AD ∥ BE.
Proof: In Quadrilateral ABCD,
AB ∥ DE [∵ Given]
AD ∥ BE [∵ By construction]
Hence, ABED is a Parallelogram.
AD = BE [∵ opposite sides of a parallelogram are equal]
AD = BC [∵ Given]
⇒ BE = BC
In ΔEBC, BE = BC [∵ Proved above]
Hence, ∠C = ∠2 …(1) [∵ In an isosceles triangle, the angles opposite to equal sides are equal]
∠A = ∠1 …(2) [∵ Opposite angles of a parallelogram are equal]
Here, ∠1 + ∠2 = 180° [∵ Linear pair]
⇒ ∠A + ∠C = 180° [∵ From the equation (1) and (2)]
⇒ ABED is a cyclic quadrilateral.