If non-parallel sides of a trapezium are equal, prove that it is cyclic.
Solution:
Consider a trapezium ABCD
Given, the non parallel sides AD and BC are equal
We have to prove that the trapezium ABCD is cyclic.
Consider a point E on BC
Join BE such that BE || AD
Now, AB || DE
AD || BE
We know that the opposite sides of a parallelogram are parallel and congruent.
So, ABED is a parallelogram.
We know that the opposite angles of a parallelogram are equal.
So, ∠BAD = ∠BED ------------------------- (1)
We know that the opposite sides of a parallelogram are equal.
So, AD = BE --------------------------------- (2)
Given, AD = BC ----------------------------- (3)
From (2) and (3),
BE = BC
We know that the angles opposite to the equal sides are equal.
∠BEC = ∠BCE -------------------------------- (4)
We know that the linear pair of angles is equal to 180 degrees.
∠BEC + ∠BED = 180°
From (1) and (4),
∠BEC + ∠BAD = 180°
We know that the sum of opposite angles in a cyclic quadrilateral is equal to 180 degrees.
Therefore, ABCD is a cyclic trapezium.
✦ Try This: Two chords AB and CD of a circle intersect each other at P outside the circle. If AB = 6cm, BP = 2cm and PD = 2.5cm, find CD.
☛ Also Check: NCERT Solutions for Class 9 Maths Chapter 10
NCERT Exemplar Class 9 Maths Exercise 10.4 Problem 2
If non-parallel sides of a trapezium are equal, prove that it is cyclic.
Summary:
A trapezium is a two-dimensional quadrilateral (as it is made of four straight lines) having a pair of parallel opposite sides. If non-parallel sides of a trapezium are equal, it is proven that the trapezium is cyclic
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