Q. 373.7( 17 Votes )
If two parallel lines are intersected by a transversal, prove that the bisectors of the two pairs of interior angles enclose a rectangle.
Given: Two lines AB and CD are parallel with transversal EF intersecting AB at M and CD at O. Bisectors of consecutive interior angles meet at N and P respectively.
To Prove: Quadrilateral MNOP is a rectangle.
Proof: Since MP, OP, MN and ON are angle bisectors
∴ ∠1 = ∠2
∠3 = ∠4
∠5 = ∠6
∠7 = ∠8
Again AB || CD and EF is a transversal
∴ ∠BMO = ∠MOC
and ∠AMO = ∠DOM (alternate angles)
∴ ∠AMO = ∠DOM
∴ ∠3 = ∠7
But these are alternate angles of sides MN and OP.
∴ MN || OP
Similarly, MP || ON
Hence MNOP is a parallelogram.
Now ∠ BMO + ∠ MOD = 180° (Consececutive interior angles)
⇒ ∠ BMO + ∠ MOD = x 180° = 90°
But in ∠ MOP, we have ∠2 + ∠7 + ∠ P = 180°.
⇒ ∠ BMO + ∠ MOD + ∠ P = 180°
∴ 90° + ∠ P = 180°
or ∠ P = 180° - 90° = 90°
Now since MNOP is a || gm and one angle is 90°.
Hence MNOP is a rectangle.
Rate this question :
How useful is this solution?
We strive to provide quality solutions. Please rate us to serve you better.
Quiz | Lines and Angles40 mins
Quiz | Lines and Angles43 mins
Lines and Angles45 mins
Quiz | Lines and Angles45 mins
NCERT | Intersecting and Non Intersecting lines)41 mins
NCERT | Discussion on Lines And Angles45 mins
NCERT | Angle Sum Property44 mins
Smart Revision | All Types of Angles31 mins
Bonus Questions on Lines and Angles42 mins
NCERT | Lines and Angles Part - 142 mins
Try our Mini CourseMaster Important Topics in 7 DaysLearn from IITians, NITians, Doctors & Academic Experts
view all courses
Dedicated counsellor for each student
24X7 Doubt Resolution
Daily Report Card
Detailed Performance Evaluation