two parallel lines AB andAC are interested by Transversal p .show that the angle bisector of interior angle of same side of Transversal make a right angleat the point of contact angle ROS =90
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11th
Maths
Straight Lines
General Equation of a Line
Prove that if two parallel ...
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Asked on December 20, 2019 by
Farhaz Ketan
Prove that if two parallel lines are intersected by a transversal, then prove that the bisectors of the interior angles form a rectangle.
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Given: Two parallel lines AB and CD and a transversal EF intersect them at G and H respectively. GM, HM, GL and HL are the bisectors of the two pairs of interior angles.
To Prove: GMHL is a rectangle.
Proof:
∵AB∥CD
∴∠AGH=∠DHG (Alternate interior angles)
⇒
2
1
∠AGH=
2
1
∠DHG
⇒∠1=∠2
(GM & HL are bisectors of ∠AGH and ∠DHG respectively)
⇒GM∥HL
(∠1 and ∠2 from a pair of alternate interior angles and are equal)
Similarly, GL∥MH
So, GMHL is a parallelogram.
∵AB∥CD
∴∠BGH+∠DHG=180
o
(Sum of interior angles on the same side of the transversal =180
o
)
⇒
2
1
∠BGH+
2
1
∠DHG=90
o
⇒∠3+∠2=90
o
.....(3)
(GL & HL are bisectors of ∠BGH and ∠DHG respectively).
In ΔGLH,∠2+∠3+∠L=180
o
⇒90
o
+∠L=180
o
Using (3)
⇒∠L=180
o
−90
o
⇒∠L=90
o
Thus, in parallelogram GMHL, ∠L=90
o
Hence, GMHL is a rectangle.
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I didn't understand sorry!!