If a pair of opposite sides of a cyclic quadrilateral are equal,prove that its diagonals are also equal.

Vedclass pdf generator app on play store
Vedclass iOS app on app store
(N/A) $ABCD$ is a cyclic quadrilateral in which one pair of opposite sides $AB = DC$. We have to prove that diagonal $AC = $ diagonal $BD$.
In $\Delta AOB$ and $\Delta DOC$,we have:
$\angle 1 = \angle 3$ [Angles in the same segment of the circle are equal]
$AB = DC$ [Given]
$\angle 2 = \angle 4$ [Angles in the same segment of the circle are equal]
Therefore,$\Delta AOB \cong \Delta DOC$ [By $ASA$ congruence rule]
Therefore,$AO = OD$ [$CPCT$] $...(1)$
And $OC = BO$ [$CPCT$] $...(2)$
Now,adding $(1)$ and $(2)$,we get:
$AO + OC = BO + OD$
$\Rightarrow AC = BD$
Hence,proved.

Explore More

Similar Questions

Prove that equal chords of a circle subtend equal angles at the centre.

State whether each of the following statements is true or false:
$(1)$ There are three circles passing through three given non-collinear points.
$(2)$ Infinitely many circles pass through two given points.

The region between an arc and the two radii,joining the centre to the end points of the arc is called a ........... .

In a cyclic quadrilateral $ABCD$,if $\angle A = \angle C$,then $\angle A = $ ........... (in $^{\circ}$)

If two chords $AB$ and $CD$ of a circle intersect at right angles (see Fig.),prove that $\text{arc } CXA + \text{arc } DZB = \text{arc } AYD + \text{arc } BWC = \text{semicircle}$.

Difficult
View Solution

Vedclass Products

For Students

Vedclass Test Series

Mock tests in real JEE/NEET style with performance analysis. 5-day free trial.

Start Free Trial
For Teachers

Exam Paper Generator

Generate Set A/B/C/D exam papers from 7.5L+ questions in 2 minutes. 3 chapters free.

Try Free
For Institutes

Online Exam Module

Live online exams with unlimited students, 360° analytics & white-label branding.

See Demo