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

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

Grade:9

4 Answers

Pranjal Animesh
21 Points
6 years ago
In quadrilateral ABCD , 
    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,
                       AB = AD
                      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
6 years ago
In quadrilateral ABCD , 
    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,
                       AB = AD
                      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
3 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,
 
                       AB = AD
 
                      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
3 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
AB=AD ( proved)
∠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
 
 
 

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