Question

In figure 2.28, line PS is a transversal
of parallel line AB and line CD. If Ray
QX, ray QY, ray RX, ray RY are angle
bisectors, then prove that QXRY is a
rectangle.​

Answer 1

Given :

Two parallel lines AB and CD are intersected by a Transversal PR in points Q and R respectively.

The bisectors of two pairs of interior angles intersect in Y and X.

To prove:

QXRY is a rectangle.

Proof :

$AB \parallel CD$

and a Transversal QR intersects them

< AQR = <QRD

( Alternate interior angles )

$\implies \frac{1}{2} \angle {AQR } = \frac{1}{2} \angle {QRD}$

( Halves of equals are equal )

=> <1 = <2

But these form a pair of equal alternate interior angles.

$\therefore QX \parallel RY \: ---(1)$

Similarly , we can show that

$RX \parallel QY \: ---(2)$

/* From (1) and (2) , */

QYRX is a parallelogram.

Now, The sum of consecutive interior angles on the same side of a Transversal is 180° .

$\angle {BQR} + \angle {QRD} = 180\degree$

$\implies \frac{1}{2} \angle {BQR} +\frac{1}{2} \angle {QRD} =\frac{1}{2} \times 180\degree$

$\implies \angle 3 + \angle 2 = 90\degree \: --(3)$

In ∆QRY ,

$\angle 3 + \angle 2 + \angle {QRY} = 180\degree$

/* Angle sum Property */

$\implies 90\degree + \angle {QRY} = 180\degree \: [ From \: (3) ]$

$\angle {QRY} = 90\degree$

$\implies QYRX \: is \:a \: rectangle .$

/* A parallelogram with one of its angles of measure 90° is a rectangle. */

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Answer 2

Given :

Two parallel lines AB and CD are intersected by a Transversal PR in points Q and R respectively.

The bisectors of two pairs of interior angles intersect in Y and X.

To prove:

QXRY is a rectangle.

Proof :

and a Transversal QR intersects them

< AQR = <QRD

( Alternate interior angles )

( Halves of equals are equal )

=> <1 = <2

But these form a pair of equal alternate interior angles.

Similarly , we can show that

From (1) and (2) ,

QYRX is a parallelogram.

Now, The sum of consecutive interior angles on the same side of a Transversal is 180° .

In ∆QRY ,

/Angle sum Property.

A parallelogram with one of its angles of measure 90° is a rectangle.