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If from any point O on a line PQ, two lines OR and OS are drawn in the opposite sides of PQ , such that angle POR=angle QOS, then prove that OR and OS lie in a line.?
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If from any point O on a line PQ, two lines OR and OS are drawn in the...
Proof that OR and OS lie in a line
Given:
- Point O lies on line PQ
- Two lines OR and OS are drawn from point O on opposite sides of line PQ
- Angle POR = Angle QOS
Proof:
1. Draw a figure:
- Draw line PQ with point O on it
- Draw lines OR and OS from point O on opposite sides of PQ
- Mark angles POR and QOS
2. Given angle POR = angle QOS:
- This implies that the lines OR and OS are mirror images of each other with respect to line PQ
3. Consider the angles formed:
- Angles POR and QOS are vertically opposite angles and hence equal
- This implies that line OR is parallel to line OS
4. Using the property of parallel lines:
- When a transversal intersects two parallel lines, the alternate interior angles are equal
- Therefore, angles POS and ROQ are equal
5. Conclusion:
- Since angles POS and ROQ are equal, it can be concluded that lines OR and OS lie on the same line
Therefore, it is proved that lines OR and OS lie in a line when angle POR is equal to angle QOS.
Given:
- Point O lies on line PQ
- Two lines OR and OS are drawn from point O on opposite sides of line PQ
- Angle POR = Angle QOS
Proof:
1. Draw a figure:
- Draw line PQ with point O on it
- Draw lines OR and OS from point O on opposite sides of PQ
- Mark angles POR and QOS
2. Given angle POR = angle QOS:
- This implies that the lines OR and OS are mirror images of each other with respect to line PQ
3. Consider the angles formed:
- Angles POR and QOS are vertically opposite angles and hence equal
- This implies that line OR is parallel to line OS
4. Using the property of parallel lines:
- When a transversal intersects two parallel lines, the alternate interior angles are equal
- Therefore, angles POS and ROQ are equal
5. Conclusion:
- Since angles POS and ROQ are equal, it can be concluded that lines OR and OS lie on the same line
Therefore, it is proved that lines OR and OS lie in a line when angle POR is equal to angle QOS.
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If from any point O on a line PQ, two lines OR and OS are drawn in the opposite sides of PQ , such that angle POR=angle QOS, then prove that OR and OS lie in a line.?
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If from any point O on a line PQ, two lines OR and OS are drawn in the opposite sides of PQ , such that angle POR=angle QOS, then prove that OR and OS lie in a line.? for UPSC 2024 is part of UPSC preparation. The Question and answers have been prepared according to the UPSC exam syllabus. Information about If from any point O on a line PQ, two lines OR and OS are drawn in the opposite sides of PQ , such that angle POR=angle QOS, then prove that OR and OS lie in a line.? covers all topics & solutions for UPSC 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for If from any point O on a line PQ, two lines OR and OS are drawn in the opposite sides of PQ , such that angle POR=angle QOS, then prove that OR and OS lie in a line.?.
If from any point O on a line PQ, two lines OR and OS are drawn in the opposite sides of PQ , such that angle POR=angle QOS, then prove that OR and OS lie in a line.? for UPSC 2024 is part of UPSC preparation. The Question and answers have been prepared according to the UPSC exam syllabus. Information about If from any point O on a line PQ, two lines OR and OS are drawn in the opposite sides of PQ , such that angle POR=angle QOS, then prove that OR and OS lie in a line.? covers all topics & solutions for UPSC 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for If from any point O on a line PQ, two lines OR and OS are drawn in the opposite sides of PQ , such that angle POR=angle QOS, then prove that OR and OS lie in a line.?.
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