# ABCD is a cyclic quadrilateral. Find all the angles of this quadrilateral.

Answer

Verified

279.9k+ 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

Define absolute refractive index of a medium

Find out what do the algal bloom and redtides sign class 10 biology CBSE

Prove that the function fleft x right xn is continuous class 12 maths CBSE

Find the values of other five trigonometric functions class 10 maths CBSE

Find the values of other five trigonometric ratios class 10 maths CBSE

Find the values of other five trigonometric functions class 10 maths CBSE

Trending doubts

Difference Between Plant Cell and Animal Cell

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

Difference between Prokaryotic cell and Eukaryotic class 11 biology CBSE

What is the past tense of read class 10 english CBSE

Change the following sentences into negative and interrogative class 10 english CBSE

What is pollution? How many types of pollution? Define it

Give 10 examples for herbs , shrubs , climbers , creepers

Which state has the longest coastline in India A Tamil class 10 social science CBSE

Write an application to the principal requesting five class 10 english CBSE