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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.?
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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.
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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.?
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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.?.
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