Answer
405k+ views
Hint: Let us suppose the quadrilateral ABCD with diagonal bisect at point O . So first let us prove that the triangle AOD and triangle COB is congruent and triangle DOC and triangle BOA is congruent by the SAS congruence rule . So from this Angle OAD and Angle OCB are alternate angles and are equal and Angle ABO and Angle CDO are alternate angles and are equal using this we can prove this
Complete step-by-step answer:
As it is given that the ABCD is an quadrilateral with it diagonal AC and BD which intersect at the point O , and It is also given that the diagonal bisect each other , that is
OA=OC and OB=OD
In triangle AOD and COB
OA=OC It is given in the question
Angle AOD = Angle COB because it is vertical opposite
OD=OB It is given in the question
Hence the triangle AOD and triangle COB is congruent by the SAS congruence rule .
So from this we can say that angle OAD = angle OCB
Similarly, we for triangle AOB and the triangle COD
OD=OB It is given in the question
Angle DOC = Angle BOA because it is vertical opposite
OA = OC It is given in the question
Hence the triangle DOC and triangle BOA is congruent by the SAS congruence rule .
So from this we can say that angle ODC = angle OBA
For lines AB and CD and the transversal line BD,
Angle ABO and Angle CDO are alternate angles and are equal.
hence both Lines are parallel ${\text{AB}}\left\| {{\text{CD}}} \right.$
For lines AD and BC, and the transversal line AC,
Angle OAD and Angle OCB are alternate angles and are equal.
hence both Lines are parallel ${\text{AD}}\left\| {{\text{BC}}} \right.$
Thus, in quadrilateral ABCD, both pairs of opposite sides are parallel.
Hence ABCD is a parallelogram.
Note: There are five ways to find out that two triangles are congruent are SSS, SAS, ASA, AAS and HL ( hypotenuse ,leg
ABC is a right-angled triangle and O is the midpoint of the side opposite to the right angle then O is equidistant from A, B, and C
Complete step-by-step answer:
![seo images](https://www.vedantu.com/question-sets/32472c7c-1fbf-4104-acec-a654fa566ba1112970607213865684.png)
As it is given that the ABCD is an quadrilateral with it diagonal AC and BD which intersect at the point O , and It is also given that the diagonal bisect each other , that is
OA=OC and OB=OD
In triangle AOD and COB
OA=OC It is given in the question
Angle AOD = Angle COB because it is vertical opposite
OD=OB It is given in the question
Hence the triangle AOD and triangle COB is congruent by the SAS congruence rule .
So from this we can say that angle OAD = angle OCB
Similarly, we for triangle AOB and the triangle COD
OD=OB It is given in the question
Angle DOC = Angle BOA because it is vertical opposite
OA = OC It is given in the question
Hence the triangle DOC and triangle BOA is congruent by the SAS congruence rule .
So from this we can say that angle ODC = angle OBA
For lines AB and CD and the transversal line BD,
Angle ABO and Angle CDO are alternate angles and are equal.
hence both Lines are parallel ${\text{AB}}\left\| {{\text{CD}}} \right.$
For lines AD and BC, and the transversal line AC,
Angle OAD and Angle OCB are alternate angles and are equal.
hence both Lines are parallel ${\text{AD}}\left\| {{\text{BC}}} \right.$
Thus, in quadrilateral ABCD, both pairs of opposite sides are parallel.
Hence ABCD is a parallelogram.
Note: There are five ways to find out that two triangles are congruent are SSS, SAS, ASA, AAS and HL ( hypotenuse ,leg
ABC is a right-angled triangle and O is the midpoint of the side opposite to the right angle then O is equidistant from A, B, and C
Recently Updated Pages
How many sigma and pi bonds are present in HCequiv class 11 chemistry CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Why Are Noble Gases NonReactive class 11 chemistry CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Let X and Y be the sets of all positive divisors of class 11 maths CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Let x and y be 2 real numbers which satisfy the equations class 11 maths CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Let x 4log 2sqrt 9k 1 + 7 and y dfrac132log 2sqrt5 class 11 maths CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Let x22ax+b20 and x22bx+a20 be two equations Then the class 11 maths CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Trending doubts
Fill the blanks with the suitable prepositions 1 The class 9 english CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
At which age domestication of animals started A Neolithic class 11 social science CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Which are the Top 10 Largest Countries of the World?
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Give 10 examples for herbs , shrubs , climbers , creepers
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Difference between Prokaryotic cell and Eukaryotic class 11 biology CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Difference Between Plant Cell and Animal Cell
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Write a letter to the principal requesting him to grant class 10 english CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Change the following sentences into negative and interrogative class 10 english CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Fill in the blanks A 1 lakh ten thousand B 1 million class 9 maths CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)