Looking at the given figure, this seems to be a very difficult question to solve. But it is exactly opposite. A very easy question to solve. All you need is have a close look at the given figure and apply following theorems.

Theorem 10.9: Angles in the same segment of a circle are equal.

Theorem 6.1: If two lines intersect each other, then the vertically opposite angles are equal.

Primarily, this entire chapter is based on the cyclic quadrilateral and its properties. To prove that the given trapezium is cyclic, we need to prove that sum of its any pair of opposite angles is 180 degrees. We have used the following very important concepts of geometry in this proof.

• Same side interior angles formed on two parallel lines are supplementary
• Perpendicular distance between two parallel lines
• RHS congruence theorem

Theorem 6.4: If a transversal intersects two parallel lines, then each pair of interior angles on the same side of the transversal is supplementary.

Theorem 7.5: (RHS congruence rule) : If in two right triangles the hypotenuse and one side of one triangle are equal to the hypotenuse and one side of the other triangle, then the two triangles are congruent.