# If in a quadrilateral abcd ,ac bisects angle a and c show that ac is perpendicular to bd

Pranjal Animesh
21 Points
5 years ago
AC and BD are diagonals . So let they intersect at O.
In triangles ABC and ADC,
angle CAB = angle CAD
angle  ACB = angle ACD
AC = AC
Therefore, triangles ABC and ADC are congruent.
NOW,
AB = AD (corresponding parts of congruent triangles)
In triangle ABD,
So, triangle ABD is isosceles.
As we know the angle bisector of the vertical angle of an isosceles triangle
is also the perpendicular bisector of the base.
So AC intersects BD at right angle.
so angle AOD = 90o.
Pranjal Animesh
21 Points
5 years ago
AC and BD are diagonals . So let they intersect at O.
In triangles ABC and ADC,
angle CAB = angle CAD
angle  ACB = angle ACD
AC = AC
Therefore, triangles ABC and ADC are congruent.
NOW,
AB = AD (corresponding parts of congruent triangles)
In triangle ABD,
So, triangle ABD is isosceles.
As we know the angle bisector of the vertical angle of an isosceles triangle
is also the perpendicular bisector of the base.
So AC intersects BD at right angle
Ayan
15 Points
2 years ago
In triangles ABC and ADC,

angle CAB = angle CAD

angle  ACB = angle ACD

AC = AC

Therefore, triangles ABC and ADC are congruent.

NOW,

AB = AD (corresponding parts of congruent triangles)

In triangle ABD,

So, triangle ABD is isosceles.

As we know the angle bisector of the vertical angle of an isosceles triangle

is also the perpendicular bisector of the base.

So AC intersects BD at right angle

Ram Kushwah
110 Points
2 years ago
Let the diagonals AC and BD intersect at O
In ΔABC and Δ ACD

∠DAC= ∠BAC
∠ACD= ∠BCA
AC is common
ΔABC Δ ACD
So AB=AD ( side opposite to equal angles)

Now In ΔAOB and Δ AOD
∠DAC= ∠BAC ( Given)
AO is common

ΔAOB Δ AOD

Thus ∠AOB= ∠AOD ( angles opposite to equal sides)

But ∠AOB are ∠AOD adjacent angles
So ∠AOB+∠AOD=180°
∠AOB+∠AOB=180°
So ∠AOB=∠AOD=90°
Hence AC is perpendicular to BD