# Two parallel lines l and m are intersected by a transversal 't'. Show that quadrilateral formed by bisectors of interior angles is a rectangle.

2
by pravin3103
quadrilateral is a parallelogram or rectangle?
rectangle

2014-12-28T11:16:00+05:30

(P & S are 2 pts on line l and Q & R are d 2 pts on line m)

It is given that PS//QR(l//m) and transversal t intersects them at point A and C respectively.

The bisectors of angle PAC and angle ACQ intersect at B and bisectors of angle ACR and SAC intersect at D.

To show that: Quadrilateral ABCD is a rectangle

Now, angle PAC= angle ACR(alternate angles as l//m and t is a transversal)

i.e angle BAC= angle ACD

These form a pair of alternate angles for lines AB and DC with AC as transversal and they are also equal

So AB//DC

Therefore quadrilateral ABCD is a //gm

Also angle PAC + angle CAS= 180 degree(linear pair)

1/2 angle PAC + 1/2 angle CAS= 1/2 x 180 degree

or angle BAC + angle CAD = 90 degree

So ABCD is a //gm in which 1 angle is 90 degree

Therefore ABCD is a rectangle.

if u give a diagram that will be good. please give diagram
Bt I hv mentioned d construction that would help in getting d diagram.
2014-12-28T11:50:44+05:30

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LL and MM are parallel lines.  tt is a transverse line.  They intersect at A and D.    The two angular bisectors for the acute and obtuse angles are drawn. They intersect at B and C.

We have to prove that ABCD is a rectangle.

Angle BAD = angle  BAL  (b3 is bisector)
angle DAC = angle CAL  (b1 is bisector)
angles LAB + angle BAD + Angle DAC + angle CAL = 180 deg
=> 2 (angle BAD + angle DAC) = 180 deg
Hence angle BAC = 90 deg

Similarly,  angle BDC = 90 deg.