Math, asked by branstongly, 9 months ago

ABCD is a quadrilateral such that diagonal AC bisects the angles ∠A and ∠C. Prove that AB = AD and CB = CD.

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Answered by lekshikar
9

Answer:

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.

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Read more on Brainly.in - https://brainly.in/question/3868163#readmoreAC 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.

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