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the sides ab and ac of a triangle ABC are produced to P and Q respectively. if the bisector of angle abc and angle ACB intersected at then prove that is equal to angle BOC is equal to 90 minus half angle A
? Related: Hots Questions- Triangles
Most Upvoted Answer
the sides ab and ac of a triangle ABC are produced to P and Q respecti...
**Given:**
- Triangle ABC with sides AB and AC produced to points P and Q respectively.
- The bisector of angle ABC and angle ACB intersect at point O.
**To prove:**
- Angle AOB = Angle BOC = 90 - (1/2) * Angle A
**Proof:**
**1. Construction:**
- Join OP and OQ.
**2. Statement:**
- Angle BOP = Angle COQ (Vertically opposite angles)
- Angle AOB = Angle COQ + Angle COB (Angle addition property)
**3. Proof:**
- Since OP is the bisector of angle ABC, it divides angle ABC into two equal angles, i.e., Angle BOP = Angle POC.
- Similarly, since OQ is the bisector of angle ACB, it divides angle ACB into two equal angles, i.e., Angle COQ = Angle QOB.
- Therefore, Angle BOP = Angle POC = Angle COQ = Angle QOB.
- By vertically opposite angles, Angle BOP = Angle COQ.
- Using the angle addition property, Angle AOB = Angle COQ + Angle COB.
- Substitute Angle BOP = Angle COQ, Angle POC = Angle BOC, and Angle QOB = Angle BOC to get Angle AOB = Angle COQ + Angle COB = Angle BOP + Angle BOC.
- Therefore, Angle AOB = Angle BOP + Angle BOC.
- Since Angle BOP = Angle BOC, we can rewrite the equation as Angle AOB = Angle BOC + Angle BOC = 2 * Angle BOC.
- Rearrange the equation to get Angle BOC = (1/2) * Angle AOB.
- Since Angle AOB = 180 - Angle A (sum of angles in a triangle), we substitute it in the equation to get Angle BOC = (1/2) * (180 - Angle A).
- Simplify to get Angle BOC = 90 - (1/2) * Angle A.
- Therefore, Angle BOC = 90 - (1/2) * Angle A.
Hence, proved.
- Triangle ABC with sides AB and AC produced to points P and Q respectively.
- The bisector of angle ABC and angle ACB intersect at point O.
**To prove:**
- Angle AOB = Angle BOC = 90 - (1/2) * Angle A
**Proof:**
**1. Construction:**
- Join OP and OQ.
**2. Statement:**
- Angle BOP = Angle COQ (Vertically opposite angles)
- Angle AOB = Angle COQ + Angle COB (Angle addition property)
**3. Proof:**
- Since OP is the bisector of angle ABC, it divides angle ABC into two equal angles, i.e., Angle BOP = Angle POC.
- Similarly, since OQ is the bisector of angle ACB, it divides angle ACB into two equal angles, i.e., Angle COQ = Angle QOB.
- Therefore, Angle BOP = Angle POC = Angle COQ = Angle QOB.
- By vertically opposite angles, Angle BOP = Angle COQ.
- Using the angle addition property, Angle AOB = Angle COQ + Angle COB.
- Substitute Angle BOP = Angle COQ, Angle POC = Angle BOC, and Angle QOB = Angle BOC to get Angle AOB = Angle COQ + Angle COB = Angle BOP + Angle BOC.
- Therefore, Angle AOB = Angle BOP + Angle BOC.
- Since Angle BOP = Angle BOC, we can rewrite the equation as Angle AOB = Angle BOC + Angle BOC = 2 * Angle BOC.
- Rearrange the equation to get Angle BOC = (1/2) * Angle AOB.
- Since Angle AOB = 180 - Angle A (sum of angles in a triangle), we substitute it in the equation to get Angle BOC = (1/2) * (180 - Angle A).
- Simplify to get Angle BOC = 90 - (1/2) * Angle A.
- Therefore, Angle BOC = 90 - (1/2) * Angle A.
Hence, proved.
Community Answer
the sides ab and ac of a triangle ABC are produced to P and Q respecti...
a+b+c=180b+c=180-a ...(1)
obc=1/2abc
ocb=1/2acb
obc+ocb=1/2(abc+acb)
using (1)
obc+ocb=90-1/2a ...(2)
but, boc=180-(obc+ocb)
so using (2) boc=180-(90-1/2a)
==>boc= 90+12/a
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the sides ab and ac of a triangle ABC are produced to P and Q respectively. if the bisector of angle abc and angle ACB intersected at then prove that is equal to angle BOC is equal to 90 minus half angle A Related: Hots Questions- Triangles?
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the sides ab and ac of a triangle ABC are produced to P and Q respectively. if the bisector of angle abc and angle ACB intersected at then prove that is equal to angle BOC is equal to 90 minus half angle A Related: Hots Questions- Triangles? 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 the sides ab and ac of a triangle ABC are produced to P and Q respectively. if the bisector of angle abc and angle ACB intersected at then prove that is equal to angle BOC is equal to 90 minus half angle A Related: Hots Questions- Triangles? covers all topics & solutions for Class 9 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for the sides ab and ac of a triangle ABC are produced to P and Q respectively. if the bisector of angle abc and angle ACB intersected at then prove that is equal to angle BOC is equal to 90 minus half angle A Related: Hots Questions- Triangles?.
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