Question
$\text{ABCD}$ ia a cyclic quadrilateral in which $BA$ and $CD$ when produced meet in $E$ and $EA = ED$. Prove that:
$i. AD \| BC$.
$ii. EB = EC$.
$i. AD \| BC$.
$ii. EB = EC$.
✓
Answer
Given $ABCD$ is a cyclic quadrilateral in which $EA = ED$ To prove:
$i. AD \| BC$.
$ii. EB = EC$.
Proof:
$i.$ Since $EA = ED$
Then, $\angle\text{EAD}=\angle\text{EDA}\dots(1) [$Oppo. angles to equal sides$]$
Since, $\text{ABCD}$ is a cyclic quadrilateral
Then, $\angle\text{ABC}+\angle\text{ADC}=180^\circ$
But $\angle\text{ABC}+\angle\text{EBC}=180^\circ [$Linear pair of angles$]$
Then, $\angle\text{ADC}=\angle\text{EBC}\dots(2)$
Compare equations $(1)$ and $(2)$
$\angle\text{EAD}=\angle\text{EBC}\dots(3)$
Since, corresponding angles are equal
Then, $BC \| AD$
$ii.$ From equation $(3)$
$\angle\text{EAD}=\angle\text{EBC}\dots(3)$
Similarly $\angle\text{EDA}=\angle\text{ECB}\dots(4)$
Compare equations $(1)(3)$ and $(4)$
$\angle\text{EBC}=\angle\text{ECB}$
$\Rightarrow \text{EB}=\text{EC} [$Opposite angles to equal sides$]$
Need a full question paper?
Generate a complete, print-ready paper with questions like this in minutes — across 16+ boards, with answer keys.
Explore more
Similar questions
If the area of an equilateral triangle is $81\sqrt{3}\text{cm}^2,$ find its height.
→If $\text{x}=\frac{1}{2-\sqrt{3}},$ find the value of $x^3 - 2x^2 - 7x + 5$.
→A well with $14\ m$ diameter is dug $8\ m$ deep. The earth taken out of it has been evenly spread all around it to a width of $21\ m$ to form an embankment. Find the height of the embankment.
→In a cyclic quadrilateral ABCD if $\text{m}\angle\text{A}=\big(\text{m}\angle\text{C}\big).$ Find $\text{m}\angle\text{A}.$
→In a rhombus ABCD, the altitude from D to the side AB bisects AB. Find the angles of the rhombus.
$ABC$ is a Triangle. $D$ is a point on Ab such that $\text{AD}=\frac{1}{4}\text{AD}$ and $E$ is a point on $AC$ such that $\text{AE}=\frac{1}{4}\text{AC.}$ prove that $\text{DE}=\frac{1}{4}\text{BC}.$
→Find the length of a chord which is at a distance of 4cm from the centre of a circle of radius 6cm.
Find the area of a triangle two sides of which are 18cm and 10cm and the perimeter is 42cm.
Draw an angle of $80^\circ$ with the help of a protractor. Then construct angles of :
$i. 40^\circ$
$ii. 160^\circ$
$iii. 120^\circ$
→$i. 40^\circ$
$ii. 160^\circ$
$iii. 120^\circ$
Two lines are respectively perpendicular to two parallel lines. Show that they are parallel to each other.