In an isosceles trapezium AB is parallel to DC and the alternate interior angles are supplementary. So, the angles ∠A and ∠D are supplementary.which means 
∠A + ∠D = 180

Similarly, ∠B and ∠C are supplementary.
so,  ∠B + ∠C = 180
Since the trapezium is the isosceles trapezium, the base angles are equal. ∠C = ∠DSo by subsituting the anles we get
∠A + ∠C = 180
∠B + ∠D = 180 

so the opposite angles are also supplementary.

Isosceles Trapezium:
An isosceles trapezium is a quadrilateral with two parallel sides of equal length. This means that the two non-parallel sides are unequal in length. The diagonals of an isosceles trapezium are also unequal in length.

Opposite Angles:
In an isosceles trapezium, the opposite angles are formed by the intersection of the diagonals. Let's call the trapezium ABCD, where AB and CD are the parallel sides, and AD and BC are the non-parallel sides. The diagonals of the trapezium intersect at point O.

Proof:
To prove that the opposite angles in an isosceles trapezium are supplementary, we will use the property of the transversal line intersecting two parallel lines.

Step 1: Draw the trapezium and diagonals
Draw an isosceles trapezium ABCD and its diagonals AC and BD.

Step 2: Identify the angles
Label the angles formed by the intersection of the diagonals as ∠A, ∠B, ∠C, and ∠D.

Step 3: Identify the parallel sides
Identify the parallel sides of the trapezium, which are AB and CD.

Step 4: Identify the transversal line
The diagonals AC and BD act as transversal lines intersecting the parallel sides AB and CD.

Step 5: Apply the property of transversal lines
According to the property of the transversal line intersecting two parallel lines, the alternate interior angles are equal.

Therefore, ∠A = ∠C and ∠B = ∠D.

Step 6: Prove the supplementary property
To show that the opposite angles are supplementary, we need to prove that ∠A + ∠B = 180° and ∠C + ∠D = 180°.

Using the property stated in Step 5, we can substitute the angles:

∠A + ∠B = ∠C + ∠D

Since ∠A = ∠C and ∠B = ∠D:

∠A + ∠B = ∠A + ∠B

This equation is always true, which means that the opposite angles in an isosceles trapezium are supplementary.

Conclusion:
In an isosceles trapezium, the opposite angles formed by the intersection of the diagonals are supplementary. This is proved using the property of the transversal line intersecting two parallel lines and the fact that the angles formed by the intersection of the diagonals are equal.