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Grade: 9
        If in a quadrilateral abcd ,ac bisects angle a and c show that ac is perpendicular to bd
10 months ago

Answers : (2)

Pranjal Animesh
21 Points
							
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.
9 months ago
Pranjal Animesh
21 Points
							
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
9 months ago
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