Answer

Verified

470.1k+ views

Consider trapezium ABCD such that AB is parallel to CD and $AD = BC$.

Consider two points M and N that can be considered as the foot of perpendicular drawn on AB from vertices D and C respectively.

Now, if we compare $\Delta DAM$and $\Delta CBN$, we have:

$AD = BC{\text{ }}...{\text{(Given)}}$

$\angle AMD = \angle BNC{\text{ }}...{\text{(Right angles i}}{\text{.e}}{\text{. 9}}{{\text{0}}^ \circ }{\text{)}}$

And $DM = CN{\text{ }}....{\text{(Distance between two parallel lines)}}$

From this we can say that both the triangles are congruent.

$\Delta DAM \cong \Delta CBN$

We know that corresponding parts of congruent triangles are equal. So we have:

$ \Rightarrow \angle A = \angle B .....(i)$

Also $\angle B + \angle C = {180^ \circ }{\text{ }}...{\text{(sum of co - interior angles)}}$

Substituting the value of $\angle B$ from equation first, we have:

$ \Rightarrow \angle A + \angle C = {180^ \circ } ....(ii)$

Equation $(ii)$ shows that the sum of the pair of opposite angles of trapezium ABCD is ${180^ \circ }$.

Therefore the trapezium is a cyclic quadrilateral.

Note: A quadrilateral is said to be cyclic quadrilateral if all of its 4 vertices lie on the same circle.

Recently Updated Pages

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

A Paragraph on Pollution in about 100-150 Words

One cusec is equal to how many liters class 8 maths CBSE

The Equation xxx + 2 is Satisfied when x is Equal to Class 10 Maths

What were the social economic and political conditions class 10 social science CBSE

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

Trending doubts

The highest dam in India is A Bhakra dam B Tehri dam class 10 social science CBSE

Write all prime numbers between 80 and 100 class 8 maths CBSE

Who administers the oath of office to the President class 10 social science CBSE

The first paper mill in India was established in a class 10 social science CBSE

The Equation xxx + 2 is Satisfied when x is Equal to Class 10 Maths

Why is there a time difference of about 5 hours between class 10 social science CBSE

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

10 examples of evaporation in daily life with explanations

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