ABCD is a quadrilateral - OCR - GCSE Maths - Question 10 - 2017 - Paper 1

In triangle ABD, since AD = AB, we have

$\angle ADB = \angle ABD$.

Similarly, in triangle CDB, since CD = CB, we conclude that

$\angle DCB = \angle CDB.$

From the isosceles triangle properties, we can express that

$\angle ADB + \angle DCB = \angle ADC$.

And since

$\angle ABD + \angle CDB = \angle ABC$,

we realize that both pairs of angles are equal.

Thus, by the properties of the angles in isosceles triangles, we have

$\angle ADC = \angle ABC$.

This completes the proof.