If two parallel lines are intersected by a transversal, then the bisectors of the two pairs of interior angles enclose:-Required to answer. Single choice.
(1 Point)
(a) a Square
(b) a Rectangle
(c) a Parallelogram
(d) a Trapezium
Answers
Hey mate,
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)
⇒ 21∠AGH=21∠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=180o
(Sum of interior angles on the same side of the transversal =180o )
⇒ 21∠BGH+21∠DHG=90o
⇒∠3+∠2=90o
.....(3)
(GL & HL are bisectors of ∠BGH and ∠DHG respectively).
In ΔGLH,∠2+∠3+∠L=180o
⇒90o+∠L=180 oUsing (3)
⇒∠L=180 o −90 o
⇒∠L=90 o
Thus, in parallelogram GMHL, ∠L=90o
Hence, GMHL is a rectangle.