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If two parallel lines are intersected by a transversal, then prove that the bisectors of any pair of alternate interior angles are parallel?
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Proof that the bisectors of any pair of alternate interior angles are parallel
Given: Two parallel lines intersected by a transversal
To prove: The bisectors of any pair of alternate interior angles are parallel
Proof:
Conclusion: Therefore, we have proved that the bisectors of any pair of alternate interior angles are parallel when two parallel lines are intersected by a transversal.
Given: Two parallel lines intersected by a transversal
To prove: The bisectors of any pair of alternate interior angles are parallel
Proof:
- Let AB and CD be two parallel lines intersected by a transversal EF at point G.
- Let H be a point on line EF, and let angles AHD and BGC be alternate interior angles.
- Let AI be the bisector of angle AHD and BJ be the bisector of angle BGC.
- Let JI intersect line EF at point K.
- Since AI bisects angle AHD, we have angle AHI = angle IHD.
- Similarly, since BJ bisects angle BGC, we have angle BJG = angle JGC.
- Since AB and CD are parallel lines, we have angle AHI = angle BJG (alternate interior angles).
- Therefore, angle IHD = angle JGC.
- Since angles IHD and JGC are corresponding angles formed by parallel lines, they are equal.
- Therefore, angles JKI and IKH are equal.
- Since the sum of angles in a triangle is 180 degrees, we have angle JKG + angle GKI + angle IKH = 180 degrees.
- Substituting angle JGC for angle AHI, we have angle JKG + angle GKI + angle JGC = 180 degrees.
- Therefore, angle JKG + angle GKI + angle JKI = 180 degrees.
- Since angles JKI and IKH are equal, we have angle JKG + angle GKI + angle IKH = 180 degrees.
- Therefore, angle JKG + angle GKI + angle JKG = 180 degrees.
- Therefore, the sum of angles JKG and GKI is equal to 180 degrees.
- Therefore, the lines JI and EF are parallel.
- Therefore, the bisectors of any pair of alternate interior angles are parallel.
Conclusion: Therefore, we have proved that the bisectors of any pair of alternate interior angles are parallel when two parallel lines are intersected by a transversal.
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If two parallel lines are intersected by a transversal, then prove that the bisectors of any pair of alternate interior angles are parallel?
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If two parallel lines are intersected by a transversal, then prove that the bisectors of any pair of alternate interior angles are parallel? 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 If two parallel lines are intersected by a transversal, then prove that the bisectors of any pair of alternate interior angles are parallel? covers all topics & solutions for Class 9 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for If two parallel lines are intersected by a transversal, then prove that the bisectors of any pair of alternate interior angles are parallel?.
If two parallel lines are intersected by a transversal, then prove that the bisectors of any pair of alternate interior angles are parallel? 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 If two parallel lines are intersected by a transversal, then prove that the bisectors of any pair of alternate interior angles are parallel? covers all topics & solutions for Class 9 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for If two parallel lines are intersected by a transversal, then prove that the bisectors of any pair of alternate interior angles are parallel?.
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