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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 insid...
Understanding the Given Quadrilateral
To prove that BC || AD in quadrilateral ABCD, we start by clarifying the conditions provided:
- AB = CD
- OA = OD (from point O to points A and D)
- OB = OC (from point O to points B and C)
Applying the Properties of Cyclic Quadrilaterals
1. Cyclic Quadrilateral Concept: Since OA = OD and OB = OC, we can infer that triangles OAB and OCD are congruent. This is because:
- OA = OD (given)
- OB = OC (given)
- AB = CD (given)
2. Angle Relationships: The congruence of triangles OAB and OCD implies that:
- Angle OAB = Angle OCD
- Angle OBA = Angle ODC
Using Alternate Interior Angles
3. Transversal Implication: Now, we analyze lines AD and BC. Since angles OAB and OCD are equal, we can deduce:
- If a line (like AD) intersects another line (like BC), and the interior angles formed are equal, then by the converse of the alternate interior angle theorem, AD is parallel to BC.
Conclusion: Establishing Parallel Lines
4. Final Statement: Therefore, given the conditions of the quadrilateral and the properties derived from the angles, we conclude that:
- BC is parallel to AD (BC || AD).
This completes the proof that in quadrilateral ABCD, with the given properties, BC is indeed parallel to AD.
To prove that BC || AD in quadrilateral ABCD, we start by clarifying the conditions provided:
- AB = CD
- OA = OD (from point O to points A and D)
- OB = OC (from point O to points B and C)
Applying the Properties of Cyclic Quadrilaterals
1. Cyclic Quadrilateral Concept: Since OA = OD and OB = OC, we can infer that triangles OAB and OCD are congruent. This is because:
- OA = OD (given)
- OB = OC (given)
- AB = CD (given)
2. Angle Relationships: The congruence of triangles OAB and OCD implies that:
- Angle OAB = Angle OCD
- Angle OBA = Angle ODC
Using Alternate Interior Angles
3. Transversal Implication: Now, we analyze lines AD and BC. Since angles OAB and OCD are equal, we can deduce:
- If a line (like AD) intersects another line (like BC), and the interior angles formed are equal, then by the converse of the alternate interior angle theorem, AD is parallel to BC.
Conclusion: Establishing Parallel Lines
4. Final Statement: Therefore, given the conditions of the quadrilateral and the properties derived from the angles, we conclude that:
- BC is parallel to AD (BC || AD).
This completes the proof that in quadrilateral ABCD, with the given properties, BC is indeed 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.Answer as easy as possible?
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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.Answer as easy as possible? 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.Answer as easy as possible? 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.Answer as easy as possible?.
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.Answer as easy as possible? 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.Answer as easy as possible? 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.Answer as easy as possible?.
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