Question

21. ABCD is a rectangle in which diagonal AC bisects LA as well as _C.
Show that (i) ABCD is a square, (ii) diagonal BD bisects LB as well
as D.​

Answer 1

Given:ABCD is a rectangle. Diagonal AC bisect angle A as well as angle C.

To proof : (i)ABCD is a square.

(ii) diagonal BD bisect angle B as well as angle D

proof :-

(I) ABCD is a rectangle

angle A=angle C

1/2 angle A = 1/2angle C

angle DAC=1/2angle ( AC bisects angle and angle C)

CD=DA(sides opposite to equal angles are equal)

AB=BC=CD=DA

ABCD is a rectangle and all the sides are equal.

hence, ABCD is a square.

(ii) let us join BD

in BCD,

angle CDB=angle CBD(angles opposite to equal sides are equal).

BC=CD(sides of a square are equal to each other)

angle CDB=angle ABD( alternate inerior angles for AB //CD)

angle CDB=angle ABD

BD bisects angle B

also, angle CDB=angle ADB(alternate interior able for BC//AD)

angle CDB=angle ABD

BD bisects angle D as well as Angel B