UPSC Exam > UPSC Questions > ABCD is a quadrilateral in which AB = CD. Als...
Start Learning for Free
ABCD is a quadrilateral in which AB = CD. Also there is a point O inside the quadrilateral such that OA = OD and OB = OC. Prove that BC || AD.?
Most Upvoted Answer
ABCD is a quadrilateral in which AB = CD. Also there is a point O insi...
Given Conditions
- ABCD is a quadrilateral where AB = CD.
- Point O is inside the quadrilateral such that OA = OD and OB = OC.
Objective
- To prove that BC is parallel to AD (BC || AD).
Proof Steps
1. Triangles Involved
- Consider triangles OAB and OCD.
- Since OA = OD (given) and AB = CD (given), we can use these equalities to establish a relationship between the triangles.
2. Angles in Triangles
- In triangles OAB and OCD, we can state:
- Angle OAB = Angle OCD (since AB = CD and they are opposite sides).
- Angle OBA = Angle ODC (by similar reasoning).
3. Application of the Isosceles Triangle
- Since OB = OC (given), triangle OBC is isosceles. Thus, angles OBC = OCB.
4. Establishing Parallel Lines
- We now analyze angles formed by the transversal line AD with the lines BC and AD:
- Angle OAB + Angle OBC = Angle ODC + Angle OBA (Sum of angles at point O).
- Since angles OAB = Angle ODC and angles OBA = Angle OBC, we conclude that the sum of angles on either side is equal.
5. Conclusion
- By the converse of the alternate interior angles theorem, if the angles formed by a transversal are equal, then the lines are parallel. Hence, we conclude that BC || AD.
Final Result
- Therefore, it is proved that BC is parallel to AD.
- ABCD is a quadrilateral where AB = CD.
- Point O is inside the quadrilateral such that OA = OD and OB = OC.
Objective
- To prove that BC is parallel to AD (BC || AD).
Proof Steps
1. Triangles Involved
- Consider triangles OAB and OCD.
- Since OA = OD (given) and AB = CD (given), we can use these equalities to establish a relationship between the triangles.
2. Angles in Triangles
- In triangles OAB and OCD, we can state:
- Angle OAB = Angle OCD (since AB = CD and they are opposite sides).
- Angle OBA = Angle ODC (by similar reasoning).
3. Application of the Isosceles Triangle
- Since OB = OC (given), triangle OBC is isosceles. Thus, angles OBC = OCB.
4. Establishing Parallel Lines
- We now analyze angles formed by the transversal line AD with the lines BC and AD:
- Angle OAB + Angle OBC = Angle ODC + Angle OBA (Sum of angles at point O).
- Since angles OAB = Angle ODC and angles OBA = Angle OBC, we conclude that the sum of angles on either side is equal.
5. Conclusion
- By the converse of the alternate interior angles theorem, if the angles formed by a transversal are equal, then the lines are parallel. Hence, we conclude that BC || AD.
Final Result
- Therefore, it is proved that BC is parallel to AD.
Attention UPSC Students!
To make sure you are not studying endlessly, EduRev has designed UPSC study material, with Structured Courses, Videos, & Test Series. Plus get personalized analysis, doubt solving and improvement plans to achieve a great score in UPSC.
|
Explore Courses for UPSC exam
|
|
Similar UPSC Doubts
Top Courses for UPSCView all
ABCD is a quadrilateral in which AB = CD. Also there is a point O inside the quadrilateral such that OA = OD and OB = OC. Prove that BC || AD.?
Question Description
ABCD is a quadrilateral in which AB = CD. Also there is a point O inside the quadrilateral such that OA = OD and OB = OC. Prove that BC || AD.? for UPSC 2024 is part of UPSC preparation. The Question and answers have been prepared according to the UPSC exam syllabus. Information about ABCD is a quadrilateral in which AB = CD. Also there is a point O inside the quadrilateral such that OA = OD and OB = OC. Prove that BC || AD.? covers all topics & solutions for UPSC 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for ABCD is a quadrilateral in which AB = CD. Also there is a point O inside the quadrilateral such that OA = OD and OB = OC. Prove that BC || AD.?.
ABCD is a quadrilateral in which AB = CD. Also there is a point O inside the quadrilateral such that OA = OD and OB = OC. Prove that BC || AD.? for UPSC 2024 is part of UPSC preparation. The Question and answers have been prepared according to the UPSC exam syllabus. Information about ABCD is a quadrilateral in which AB = CD. Also there is a point O inside the quadrilateral such that OA = OD and OB = OC. Prove that BC || AD.? covers all topics & solutions for UPSC 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for ABCD is a quadrilateral in which AB = CD. Also there is a point O inside the quadrilateral such that OA = OD and OB = OC. Prove that BC || AD.?.
Solutions for ABCD is a quadrilateral in which AB = CD. Also there is a point O inside the quadrilateral such that OA = OD and OB = OC. Prove that BC || AD.? in English & in Hindi are available as part of our courses for UPSC.
Download more important topics, notes, lectures and mock test series for UPSC Exam by signing up for free.
Here you can find the meaning of ABCD is a quadrilateral in which AB = CD. Also there is a point O inside the quadrilateral such that OA = OD and OB = OC. Prove that BC || AD.? defined & explained in the simplest way possible. Besides giving the explanation of
ABCD is a quadrilateral in which AB = CD. Also there is a point O inside the quadrilateral such that OA = OD and OB = OC. Prove that BC || AD.?, a detailed solution for ABCD is a quadrilateral in which AB = CD. Also there is a point O inside the quadrilateral such that OA = OD and OB = OC. Prove that BC || AD.? has been provided alongside types of ABCD is a quadrilateral in which AB = CD. Also there is a point O inside the quadrilateral such that OA = OD and OB = OC. Prove that BC || AD.? theory, EduRev gives you an
ample number of questions to practice ABCD is a quadrilateral in which AB = CD. Also there is a point O inside the quadrilateral such that OA = OD and OB = OC. Prove that BC || AD.? tests, examples and also practice UPSC tests.
|
|
Explore Courses for UPSC exam
|
|
Signup for Free!
Signup to see your scores go up within 7 days! Learn & Practice with 1000+ FREE Notes, Videos & Tests.
|
© EduRev
|
Education Revolution
|
Follow Us
|
Signup to see your scores
go up within 7 days!
Access 1000+ FREE Docs, Videos and Tests
Takes less than 10 seconds to signup