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

To prove that angle ADC is equal to angle ABC, we can use the properties of isosceles triangles.

1. Identify Triangles: In quadrilateral ABCD, we observe that triangles ABD and CDB can be formed.

2. Properties of Triangles: Since AD = AB and CD = CB, triangles ABD and CDB are isosceles triangles.

3. Angles in Isosceles Triangle: In triangle ABD, since AD = AB, the angles opposite these sides are equal. Therefore, ( \angle ADB = \angle ABD ).

4. Similar Argument for Triangle CDB: Similarly, in triangle CDB, since CD = CB, we conclude that ( \angle CDB = \angle BCD ).

5. Sum of Angles in Quadrilateral: By the exterior angle theorem, angle ADC (which is ( \angle ADB + \angle CDB )) equals the sum of the opposite interior angles of triangles ABD and CDB.

6. Conclusion: Hence, ( \angle ADC = \angle ABC ), proving the statement as required.