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In quadrilateral ABCD, AO and BO are the bisector of angle A and B respectively. Prove that angle AOB=1/2 (angle C+D)
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In quadrilateral ABCD, AO and BO are the bisector of angle A and B res...
Proof:
We need to prove that angle AOB is half of angle CD.
Given:
- Quadrilateral ABCD
- AO and BO are the bisectors of angle A and B respectively
To Prove:
- Angle AOB is 1/2 angle CD
Proof:
- Let angle AOB be x and angle DCB be y
- Since AO and BO are bisectors of angle A and B respectively, we have:
- Angle AOC = Angle COB = (180 - x)/2
- Angle BOD = Angle DOA = (180 - y)/2
- Since the angles in a quadrilateral sum up to 360 degrees, we have:
- Angle AOC + Angle COB + Angle BOD + Angle DOA = 360
- (180 - x)/2 + (180 - x)/2 + (180 - y)/2 + (180 - y)/2 = 360
- 360 - x - y = 2x + 2y - 720
- Simplifying, we get:
- 3x + 3y = 540
- x + y = 180
- Since angle AOB is opposite to angle COD, we have:
- x + y = Angle COD
- Substituting the value of x + y, we get:
- Angle COD = 180
- Therefore, angle AOB is half of angle COD, i.e. angle AOB = 1/2 angle CD
Conclusion:
Hence, we have proved that angle AOB is half of angle CD in quadrilateral ABCD where AO and BO are bisectors of angle A and B respectively.
We need to prove that angle AOB is half of angle CD.
Given:
- Quadrilateral ABCD
- AO and BO are the bisectors of angle A and B respectively
To Prove:
- Angle AOB is 1/2 angle CD
Proof:
- Let angle AOB be x and angle DCB be y
- Since AO and BO are bisectors of angle A and B respectively, we have:
- Angle AOC = Angle COB = (180 - x)/2
- Angle BOD = Angle DOA = (180 - y)/2
- Since the angles in a quadrilateral sum up to 360 degrees, we have:
- Angle AOC + Angle COB + Angle BOD + Angle DOA = 360
- (180 - x)/2 + (180 - x)/2 + (180 - y)/2 + (180 - y)/2 = 360
- 360 - x - y = 2x + 2y - 720
- Simplifying, we get:
- 3x + 3y = 540
- x + y = 180
- Since angle AOB is opposite to angle COD, we have:
- x + y = Angle COD
- Substituting the value of x + y, we get:
- Angle COD = 180
- Therefore, angle AOB is half of angle COD, i.e. angle AOB = 1/2 angle CD
Conclusion:
Hence, we have proved that angle AOB is half of angle CD in quadrilateral ABCD where AO and BO are bisectors of angle A and B respectively.
Community Answer
In quadrilateral ABCD, AO and BO are the bisector of angle A and B res...
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In quadrilateral ABCD, AO and BO are the bisector of angle A and B respectively. Prove that angle AOB=1/2 (angle C+D)
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In quadrilateral ABCD, AO and BO are the bisector of angle A and B respectively. Prove that angle AOB=1/2 (angle C+D) for Class 9 2024 is part of Class 9 preparation. The Question and answers have been prepared according to the Class 9 exam syllabus. Information about In quadrilateral ABCD, AO and BO are the bisector of angle A and B respectively. Prove that angle AOB=1/2 (angle C+D) covers all topics & solutions for Class 9 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for In quadrilateral ABCD, AO and BO are the bisector of angle A and B respectively. Prove that angle AOB=1/2 (angle C+D).
In quadrilateral ABCD, AO and BO are the bisector of angle A and B respectively. Prove that angle AOB=1/2 (angle C+D) for Class 9 2024 is part of Class 9 preparation. The Question and answers have been prepared according to the Class 9 exam syllabus. Information about In quadrilateral ABCD, AO and BO are the bisector of angle A and B respectively. Prove that angle AOB=1/2 (angle C+D) covers all topics & solutions for Class 9 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for In quadrilateral ABCD, AO and BO are the bisector of angle A and B respectively. Prove that angle AOB=1/2 (angle C+D).
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