It is commonly said that in mathematics one question can be solved in multiple ways. This question is another good example to prove it. This question has been solved by three different methods by using the following concepts.

1. Opposite angles of a parallelogram are always congruent
2. Same side interior angles on two parallel lines are supplementary
3. Using congruence postulate

Theorem 8.4: In a parallelogram, opposite angles are equal.

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.

Congruence vs. Similarity

This question has been solved by two different methods – both are equally easy to use.

• Using properties of circle
• Using SAS triangles congruence

Angle in a semicircle is a right angle

Also, following theorem has been discussed as an alternative to prove one of the sections of this question.

Theorem 10.11: The sum of either pair of opposite angles of a cyclic quadrilateral is 180 degree.

The other approach used to solve this method is the SAS congruence rule for triangles. It is highly recommended that you must know about the different congruence postulates. These are quite frequently used in geometry. In the following video tutorial, different types of congruence postulates (axioms) are explained along with the difference between congruence and similarity.

Congruence v/s Similarity