Question

diagonal AC of quadrilateral ABCD bisects angle A and angle C. prove that AB=AD and CB=CD

Answer 1

Given :- BAC = DAC
BCA = DCA

TO PROVE :- AB = AD
CB = CD

PROOF :- IN TRIANGLE ABC AND ADC
BAC = DAC
BCA = DCA
AC = AC

HENCE., TRIANGLE ABC CONGRUENT TRIANGLE ADC

IMPLIES THAT., AB = AD
CB = CD

PROVED

Answer 2

Step-by-step explanation:

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.