ABCD is a trapezium where AB is parallel to CD, measure of ∠(A) = ∠(B) = 45°. Prove that AD = BC.

Let's solve this problem step by step:

Given:
ABCD is a trapezium with AB parallel to CD.
Angle A = Angle B = 45°.
Objective:
Prove that AD = BC.

Proof:
Trapezium Properties: Since ABCD is a trapezium, we know that one pair of opposite sides are parallel. Here, AB is parallel to CD.

Angle Properties:

Angle A = 45° (Given).
Angle B = 45° (Given).
Since AB is parallel to CD and the angles at A and B are equal, the line segments AD and BC will be symmetric. This is because both these angles are made by the transversal AD and BC with the parallel lines AB and CD.

Using the properties of parallel lines: In any trapezium, if the non-parallel sides form equal angles with the parallel sides, the non-parallel sides must be equal in length. This can be seen by considering the symmetry and the geometric properties of the trapezium.

Conclusion: Since angle A and angle B are both 45°, the non-parallel sides AD and BC must be equal in length due to symmetry, which results from the equal angles and the parallel nature of AB and CD.

Thus, we have proven that AD = BC.