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In a triangle ABC , it is given that AB =AC and the bisectors of angle B and angle C intersect each other at O . If M is the point of intersection of BO produced and AC , prove that angle MOC =angle ABC?
Verified Answer
In a triangle ABC , it is given that AB =AC and the bisectors of angle...
Given ABC is a triangle, where AB= AC and the angle bisector of B and C intersect at O.
let ∠ABC=2x therefore ∠ACB=2x [angles opposite to equal sides are equal]
∠OBC=2x/2=x [since OB is the angle bisector of angle B]
similarly ∠OCB=x
∠MOC=x+x=2x [since the exterior angle is equal to the sum of the opposite interior angles ]
therefore ∠MOC=2x
i.e. ∠MOC=∠ABC
This question is part of UPSC exam. View all Class 9 courses
Most Upvoted Answer
In a triangle ABC , it is given that AB =AC and the bisectors of angle...
In triangle ABC angle MOC =angle 1
bisected angles of B be 2 and 3 .
bisected angles of c be 5 and 4.
So as AB =AC angle B is equal to angle C. Therefore angle 2 equal to angle 5 as 2 and 3 are already equal as given in question.
Angle 1 is the exterior angle of triangle BOC. so angle 1 is equal to angle 3 plus angle 5
angle 5 is equal to angle 2 so it can be said that
angle 2 plus angle 3 equal to angle 1
bisected angles of B be 2 and 3 .
bisected angles of c be 5 and 4.
So as AB =AC angle B is equal to angle C. Therefore angle 2 equal to angle 5 as 2 and 3 are already equal as given in question.
Angle 1 is the exterior angle of triangle BOC. so angle 1 is equal to angle 3 plus angle 5
angle 5 is equal to angle 2 so it can be said that
angle 2 plus angle 3 equal to angle 1
Community Answer
In a triangle ABC , it is given that AB =AC and the bisectors of angle...
Given:
- Triangle ABC with AB = AC
- Bisectors of angle B and angle C intersect at point O
- Point M is the intersection of the extension of BO and AC
To prove:
angle MOC = angle ABC
Proof:
1. Draw the figure:
Draw a triangle ABC with AB = AC. Label the bisectors of angle B and angle C as BD and CE respectively. Let the intersection point of BD and CE be O. Extend BO to intersect AC at point M.
2. Establish the given information:
In triangle ABC, AB = AC. Therefore, angles B and C are congruent.
3. Prove triangle ABO is congruent to triangle ACO:
Since AB = AC and angle B = angle C, by the side-angle-side (SAS) congruence criterion, triangle ABO is congruent to triangle ACO.
4. Prove angle MOC = angle MOB:
Since triangle ABO is congruent to triangle ACO, angle OAC = angle OAB.
Therefore, angle MOC (angle OAC) is congruent to angle MOB (angle OAB) by the vertical angles theorem.
5. Prove triangle MOC is congruent to triangle MOB:
Since angle MOC = angle MOB and MO is common to both triangles, by the angle-side-angle (ASA) congruence criterion, triangle MOC is congruent to triangle MOB.
6. Prove angle MOC = angle ABC:
Since triangle MOC is congruent to triangle MOB, angle MOC = angle MOB.
Since triangle ABO is congruent to triangle ACO, angle ABC = angle ACB.
Therefore, angle MOC = angle ABC by the transitive property of equality.
7. Conclusion:
We have proved that in triangle ABC, with AB = AC and the bisectors of angle B and angle C intersecting at O, the angle MOC is equal to the angle ABC.
- Triangle ABC with AB = AC
- Bisectors of angle B and angle C intersect at point O
- Point M is the intersection of the extension of BO and AC
To prove:
angle MOC = angle ABC
Proof:
1. Draw the figure:
Draw a triangle ABC with AB = AC. Label the bisectors of angle B and angle C as BD and CE respectively. Let the intersection point of BD and CE be O. Extend BO to intersect AC at point M.
2. Establish the given information:
In triangle ABC, AB = AC. Therefore, angles B and C are congruent.
3. Prove triangle ABO is congruent to triangle ACO:
Since AB = AC and angle B = angle C, by the side-angle-side (SAS) congruence criterion, triangle ABO is congruent to triangle ACO.
4. Prove angle MOC = angle MOB:
Since triangle ABO is congruent to triangle ACO, angle OAC = angle OAB.
Therefore, angle MOC (angle OAC) is congruent to angle MOB (angle OAB) by the vertical angles theorem.
5. Prove triangle MOC is congruent to triangle MOB:
Since angle MOC = angle MOB and MO is common to both triangles, by the angle-side-angle (ASA) congruence criterion, triangle MOC is congruent to triangle MOB.
6. Prove angle MOC = angle ABC:
Since triangle MOC is congruent to triangle MOB, angle MOC = angle MOB.
Since triangle ABO is congruent to triangle ACO, angle ABC = angle ACB.
Therefore, angle MOC = angle ABC by the transitive property of equality.
7. Conclusion:
We have proved that in triangle ABC, with AB = AC and the bisectors of angle B and angle C intersecting at O, the angle MOC is equal to the angle ABC.
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In a triangle ABC , it is given that AB =AC and the bisectors of angle B and angle C intersect each other at O . If M is the point of intersection of BO produced and AC , prove that angle MOC =angle ABC?
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In a triangle ABC , it is given that AB =AC and the bisectors of angle B and angle C intersect each other at O . If M is the point of intersection of BO produced and AC , prove that angle MOC =angle ABC? 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 a triangle ABC , it is given that AB =AC and the bisectors of angle B and angle C intersect each other at O . If M is the point of intersection of BO produced and AC , prove that angle MOC =angle ABC? covers all topics & solutions for Class 9 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for In a triangle ABC , it is given that AB =AC and the bisectors of angle B and angle C intersect each other at O . If M is the point of intersection of BO produced and AC , prove that angle MOC =angle ABC?.
In a triangle ABC , it is given that AB =AC and the bisectors of angle B and angle C intersect each other at O . If M is the point of intersection of BO produced and AC , prove that angle MOC =angle ABC? 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 a triangle ABC , it is given that AB =AC and the bisectors of angle B and angle C intersect each other at O . If M is the point of intersection of BO produced and AC , prove that angle MOC =angle ABC? covers all topics & solutions for Class 9 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for In a triangle ABC , it is given that AB =AC and the bisectors of angle B and angle C intersect each other at O . If M is the point of intersection of BO produced and AC , prove that angle MOC =angle ABC?.
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