Answer

Verified

350.4k+ views

**Hint:**We are given angles of a cyclic quadrilateral in variables. So, we should know some basic properties of the angles of a cyclic quadrilateral. We should know that the opposite angles of a cyclic quadrilateral sum up to $180^{\circ}$. So, using that property we will construct equations and then find the answer.

**Complete step by step answer:**

A cyclic quadrilateral is a quadrilateral which has all its four vertices lying on a circle. It is also sometimes called inscribed quadrilateral. The circle which consists of all the vertices of any polygon on its circumference is known as the circum-circle or circumscribed circle.

The opposite angles of a cyclic quadrilateral sum up to $180^{\circ}$. So, after looking at the figure we can say that:

$3y-5+\left(-7x+5\right)=180$

$\implies -7x+3y=180$.........$\left(1\right)$

And

$4y+20+\left(-4x\right)=180$

$\implies -4x+4y=160$

$\implies -x+y=40$

$\implies y=x+40$

Putting in $\left(1\right)$

$-7x+3\left(x+40\right)=180$

$\implies -7x+3x+120=180$

$\implies -4x=60$

$\implies x=-15$

Then since $y=x+40$, we have:

$y=40-15=25$

Now, we find all the angles one by one:

$\angle A=4y+20=4\times 25+20=120^{\circ}$

$\angle B=3y-5=3\times 25-5=70^{\circ}$

$\angle C=-4x=-4\times -15=60^{\circ}$

$\angle D=-7x+5=-7\times -15+5=110^{\circ}$

Hence, all the angles have been found.

**Note:**You can also do the same by finding any two of the adjacent angles and then subtracting that from $180^{\circ}$. For example, say you have found the angle A to be $120^{\circ}$, then there is no need to put the value of $x$ and $y$ to find the value of the opposite angle C. You can simply subtract 120 from 180 and obtain $60^{\circ}$ as the measure of the angle C.

Recently Updated Pages

How many sigma and pi bonds are present in HCequiv class 11 chemistry CBSE

Why Are Noble Gases NonReactive class 11 chemistry CBSE

Let X and Y be the sets of all positive divisors of class 11 maths CBSE

Let x and y be 2 real numbers which satisfy the equations class 11 maths CBSE

Let x 4log 2sqrt 9k 1 + 7 and y dfrac132log 2sqrt5 class 11 maths CBSE

Let x22ax+b20 and x22bx+a20 be two equations Then the class 11 maths CBSE

Trending doubts

Establish a relation between electric current and drift class 12 physics CBSE

Guru Purnima speech in English in 100 words class 7 english CBSE

Fill the blanks with the suitable prepositions 1 The class 9 english CBSE

Difference Between Plant Cell and Animal Cell

Difference between Prokaryotic cell and Eukaryotic class 11 biology CBSE

Write a letter to the principal requesting him to grant class 10 english CBSE

Copper is not used as potentiometer wire because class 12 physics CBSE

Give 10 examples for herbs , shrubs , climbers , creepers

Fill in the blanks A 1 lakh ten thousand B 1 million class 9 maths CBSE