# ABCD is a cyclic quadrilateral whose diagonals intersect at O. If $\angle ACB=50{}^\circ$ and $\angle ABC=110{}^\circ$, find $\angle BDC$.

Last updated date: 28th Mar 2023

•

Total views: 307.5k

•

Views today: 3.87k

Answer

Verified

307.5k+ views

Hint: We can solve this problem by using the property of angle inscribed by the arc, i. e. Angles inscribed by the same arc on the circumference of a circle are always EQUAL, and the property of a triangle.

Complete step-by-step answer:

Firstly, we will write the given values,

$\angle ACB=50{}^\circ$ and $\angle ABC=110{}^\circ $……………………………….. (1)

To find the required angle we should know the key concept which is given below,

Angles inscribed by the same arc on the circumference of a circle are always EQUAL, which means, angles made by arc BC at point D and A on the circumference of the circle are equal.

\[\therefore \angle BDC=\angle BAC\]……………………………………………….. (2)

To find the value of \[\angle BAC\] consider\[\vartriangle ABC\],

As the sum of angles of a triangle is always \[180{}^\circ \]

\[\therefore \angle ABC+\angle BAC+\angle ACB=180{}^\circ \]

Put the values of (1) in the above equation,

\[\Rightarrow 110{}^\circ +\angle BAC+50{}^\circ =180{}^\circ \]

\[\Rightarrow \angle BAC=180{}^\circ -160{}^\circ \]

\[\Rightarrow \angle BAC=20{}^\circ \]

Now put the value of equation (2) in the above equation,

\[\Rightarrow \angle BDC=\angle BAC=20{}^\circ \]

\[\therefore \angle BDC=20{}^\circ \]

Therefore the value of \[\angle BDC\] is \[20{}^\circ \].

Note: In the problem they have just mentioned that diagonals are passing through point O, but it is not clear that O is the centre of the circle. If we consider YOU as center then, \[\angle DCA\] will become \[40{}^\circ \] and we will get the final answer as \[\angle BDC=50{}^\circ \] which is a wrong answer.

Complete step-by-step answer:

Firstly, we will write the given values,

$\angle ACB=50{}^\circ$ and $\angle ABC=110{}^\circ $……………………………….. (1)

To find the required angle we should know the key concept which is given below,

Angles inscribed by the same arc on the circumference of a circle are always EQUAL, which means, angles made by arc BC at point D and A on the circumference of the circle are equal.

\[\therefore \angle BDC=\angle BAC\]……………………………………………….. (2)

To find the value of \[\angle BAC\] consider\[\vartriangle ABC\],

As the sum of angles of a triangle is always \[180{}^\circ \]

\[\therefore \angle ABC+\angle BAC+\angle ACB=180{}^\circ \]

Put the values of (1) in the above equation,

\[\Rightarrow 110{}^\circ +\angle BAC+50{}^\circ =180{}^\circ \]

\[\Rightarrow \angle BAC=180{}^\circ -160{}^\circ \]

\[\Rightarrow \angle BAC=20{}^\circ \]

Now put the value of equation (2) in the above equation,

\[\Rightarrow \angle BDC=\angle BAC=20{}^\circ \]

\[\therefore \angle BDC=20{}^\circ \]

Therefore the value of \[\angle BDC\] is \[20{}^\circ \].

Note: In the problem they have just mentioned that diagonals are passing through point O, but it is not clear that O is the centre of the circle. If we consider YOU as center then, \[\angle DCA\] will become \[40{}^\circ \] and we will get the final answer as \[\angle BDC=50{}^\circ \] which is a wrong answer.

Recently Updated Pages

If a spring has a period T and is cut into the n equal class 11 physics CBSE

A planet moves around the sun in nearly circular orbit class 11 physics CBSE

In any triangle AB2 BC4 CA3 and D is the midpoint of class 11 maths JEE_Main

In a Delta ABC 2asin dfracAB+C2 is equal to IIT Screening class 11 maths JEE_Main

If in aDelta ABCangle A 45circ angle C 60circ then class 11 maths JEE_Main

If in a triangle rmABC side a sqrt 3 + 1rmcm and angle class 11 maths JEE_Main

Trending doubts

Difference Between Plant Cell and Animal Cell

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

Ray optics is valid when characteristic dimensions class 12 physics CBSE

Give 10 examples for herbs , shrubs , climbers , creepers

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

List out three methods of soil conservation

Epipetalous and syngenesious stamens occur in aSolanaceae class 11 biology CBSE

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

A Short Paragraph on our Country India