Choose whether given statement is true or false:

A line l is parallel to line m and a transversal p intersects them at X, Y respectively. Bisectors of interior angles at X and Y intersect at P and Q. Then PXQY a rectangle.

Question

Choose whether given statement is true or false:

A line l is parallel to line m and a transversal p intersects them at X, Y respectively. Bisectors of interior angles at X and Y intersect at P and Q. Then PXQY a rectangle.

Accepted Answer

Drawing the diagram using given information: Now, applying property of alternative angles, As

l∥m
   and

p
   is transversal so the alternate interior angle is equal.

∴∠DXY=∠XYA

⇒
      
       ∠DXY
      2
     
     =
      
       ∠XYA
      2

From the figure XP and YQ are bisectors. Then,

∠1=∠2

XP∥QY..................(1)
    Applying property of alternative angles:As

l∥m
   and

p
   is transversal so the alternate interior angle are equal.

∴∠CXY=∠XYB

⇒
      
       ∠CXY
      2
     
     =
      
       ∠XYB
      2

Also, XQ and YP are bisectors. Then,

∠QXP=∠PYX

XQ∥PY..................(2)
   By equation (1) and (2):PXQY is a parallelogram.Applying the property of parallel lines i.e., the interior angle on the same of transversal are supplementary.

∴∠DXY+∠XYB=
      180
      °

⇒
      
       ∠DXY
      2
     
     +
      
       ∠XYB
      2
     
     =

180
        °

2

⇒∠1+∠3=
      90
      °

In

ΔXYP

⇒∠1+∠3+∠P=
      180
      °

⇒
      90
      °
     
     +∠P=
      180
      °

⇒∠P=
      90
      °

So, all the angles are right angles. Thus, PXQY is a rectangle.Hence, the statement is true and option 1 is correct.

Choose whether given statement is true or false:

A line l is parallel to line m and a transversal p intersects them at X, Y respectively. Bisectors of interior angles at X and Y intersect at P and Q. Then PXQY a rectangle.

Best Courses for You

Detailed Solution

courses

Get Expert Academic Guidance – Connect with a Counselor Today!

best study material, now at your finger tips!

download the app

Download the App

Choose whether given statement is true or false:A line l is parallel to line m and a transversal p intersects them at X, Y respectively. Bisectors of interior angles at X and Y intersect at P and Q. Then PXQY a rectangle.

Best Courses for You

Detailed Solution

courses

Get Expert Academic Guidance – Connect with a Counselor Today!

best study material, now at your finger tips!

download the app

Download the App

Choose whether given statement is true or false:

A line l is parallel to line m and a transversal p intersects them at X, Y respectively. Bisectors of interior angles at X and Y intersect at P and Q. Then PXQY a rectangle.