Question

If two parallel lines are intersected by a transversal, prove that the bisectors of the two pairs of interior angles from a ractangles.

Answer 1

GLHM is a rectangle

Step-by-step explanation:

Given: Two parallel lines AB and CD get intersected by a transversal EF at G and H respectively. The bisectors of two pairs of interior angles intersect at L and M.

To Prove: Because AB parallel to CD and transversal EF intersects them,  

<AGH = <GHD, as they are alternate interior angles.

1/2 <AGH = 1/2<GHD, as half of equal angles are equal.

So <1 = <2 (please refer to diagram for the angles)

But as these are equal alternate interior angles, as GM is parallel to HL --- (1)

Also, HM is parallel to GL --- (2)

Hence from (1) and (2), we can see that GLHM is a parallelogram as both pairs of opposite sides are parallel.

As the sum of consecutive interior angles on the same side of a transversal is 180 degress,

<BGH + <GHD = 180  degrees

So 1/2 <BGH + 1/2 <GHD will be = 90  degrees

So <3 + <2 = 90 degrees  --- (3)

In triangle GHL, <3 +<2 + <GLH = 180 degrees as it is sum of 3 angles of triangle.

So 90 degrees + <GLH = 180 degrees

From (3), we get <GLH = 90 degrees.

Therefore, GLHM is a rectangle, as it is a parallelogram with one of its angles measuring 90 degrees.

Hence proved GLHM is a rectangle.