CameraIcon
CameraIcon
SearchIcon
MyQuestionIcon
MyQuestionIcon
1
You visited us 1 times! Enjoying our articles? Unlock Full Access!
Question

In a rhombus ABCD show that diagonal AC bisects A as well as C and diagonal BD bisects B as well as D.

Open in App
Solution

Given: A rhombus ABCD.
To prove:

Diagonal AC bisects A as well as C and diagonal BD bisects B as well as D.
Proof:
In ABC
AB = BC (Sides of rhombus are equal.)
∠44 =2 (Angles opposite to equal sides are equal.)
...(1)
Now,
ADBC
(Opposite sides of rhombus are parallel.)
AC is transversal.
So, 1 =4
(Alternate interior angles)
...(2)
From (1) and (2), we get
1 =2
Thus, AC bisects A
Similarly,
Since, ABDC and AC is transversal.
So, 2 =3
(Alternate interior angles)
...(3)
From (1) and (3), we get
4 =3
Thus, AC bisects C
Hence, AC bisects C and A
In DAB
AD = AB (Sides of rhombus are equal.)
ADB=ABD (Angles opposite to equal sides are equal.)
Now,
DCAB
(Opposite sides of rhombus are parallel.)
BD is transversal.
So, CDB=DBA
(Alternate interior angles)...(5)...(4)

From (4) and (5), we get
ADB=CDB
Thus, DB bisects D
Similarly,
Since, ADBC and BD is transversal.
So, CBD =\angle ADB \) Alternate interior angles)...(6)
From (4) and (6), we get
CBD =\angle ADB \)
Thus, BD bisects \angle B \)
Hence, BD bisects D and B


flag
Suggest Corrections
thumbs-up
5
Join BYJU'S Learning Program
similar_icon
Related Videos
thumbnail
lock
MATHEMATICS
Watch in App
Join BYJU'S Learning Program
CrossIcon