Question

$\huge\pink\underline{QuEsTiOn:}$
If l and m are two parallel lines and t is a tranversal.Show that the quadrilateral formed by the bisector of pair of interior angles is rectangle..
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Answer 1

ANSWER

Given : l∥m

Transversal p intersects l & m at A & C respectively. Bisector of ∠ PAC & ∠ QCA meet at B. And, bisector of ∠ SAC & ∠ RCA meet at D.

To prove : ABCD is a rectangle.

Proof :

We know that a rectangle is a parallelogram with one angle 90

o

.

For l∥m and transversal p

∠PAC=∠ACR

So,

2

1

∠PAC=

2

1

∠ACR

So, ∠BAC=∠ACD

For lines AB and DC with AC as transversal ∠BAC & ∠ACD are alternate angles, and they are equal.

So, AB∥DC.

Similarly, for lines BC & AD, with AC as transversal ∠BAC & ∠ACD are alternate angles, and they are equal.

So, BC∥AD.

Now, In ABCD,

AB∥DC & BC∥AD

As both pair of opposite sides are parallel, ABCD is a parallelogram.

Also, for line l,

∠PAC+∠CAS=180

o

2

1

∠PAC+

2

1

∠CAS=90

o

∠BAC+∠CAD=90

o

∠BAD=90

o

.

So, ABCD is a parallelogram in which one angle is 90

o

.

Hence, ABCD is a rectangle.

Answer 2

refer to the attached images......