Question

In the given figure, 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 `square` QXRY is a rectangle.

Answer 1

Given: AB and CD are two parallel lines which are cut by a transversal PS at the points Q and R respectively. The bisectors of the interior angles intersect at points X and Y.

To prove: Quadrilateral QXRY is a rectangle.

Proof: Since AB || CD and PS is a transversal.

∠AQR = ∠DRQ    ...(Alternate interior angles)

⇒ `1/2` ∠AQR = `1/2` ∠DRQ    ...(1)

Since QX bisects ∠AQR and RY bisects ∠DRQ, then

∠XQR = `1/2`∠AQR and ∠YRQ = `1/2`∠DRQ 

∴ from (1), we get

∠XQR = ∠YRQ

But ∠XQR and ∠YRQ are alternate interior angles formed by the transversal QR with QX and RY respectively.

∴ QX || RY     ...(Alternate angles test)

Similarly, we have RX || QY. 

Hence, in quadrilateral QXRY, we have QX || RY and RX || QY.

It is known that, a quadrilateral is a parallelogram if its opposite sides are parallel.

∴ QXRY is a parallelogram.

Since sum of the interior angles on the same side of transversal is 180^∘, then

∠BQR + ∠DRQ = 180^∘

⇒ `1/2` ∠BQR + `1/2` ∠DRQ = 90^∘    ...(2)

Since QY bisects ∠BQR and RY bisects ∠DRQ, then

∠YQR = `1/2`∠BQR and ∠YRQ = `1/2`∠DRQ

∴ from (2), we get

∠YQR + ∠YRQ =  90^∘    ...(3)

In ∆QRY, we have

∠YQR + ∠YRQ + ∠QYR = 180^∘    ...(Angle sum property of triangle)

⇒ 90^∘ + ∠QYR = 180^∘    ...[Using (3)]

⇒ ∠QYR = 180^∘ − 90^∘ 

⇒ ∠QYR = 90^∘

Since QXRY is a parallelogram, then

∠QXR = ∠QYR    ...(Opposite angles of parallolegram are equal)

⇒ ∠QXR = 90^∘     ...(∵ ∠QYR = 90^∘)

Since adjacent angles in a parallelogram are supplementary, then

∠QXR + ∠XRY = 180^∘

⇒ 90^∘ + ∠XRY = 180^∘    ...(∵ ∠QXR = 90^∘)

⇒ ∠XRY = 180^∘ − 90^∘ 

⇒ ∠XRY = 90^∘

Also, ∠XQY = ∠XRY = 90^∘   ...(Opposite angles of parallolegram are equal)

Thus, QXRY is a parallelogram in which all the interior angles are right angles.

It is known that, a rectangle is a parallolegram in which each angle is a right angle.

Hence, `square` QXRY is a rectangle.