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

While visualising the different scenarios of this question, I realised that there is something wrong in this question. This question is not correct for all possible scenarios, hence NCERT team need to fix this question.

There are three possible scenarios on the basis of the statement. There is one scenario out of three possible scenarios for which this statement is not valid.

It is request to the readers to share this question so that it reaches NCERT team and necessary changes are made.

It is bit tricky question to solve (not difficult though). Such questions are generally proved by assuming the other way and proving the assumption wrong. For example, in this question we assumed that the point of intersection does not lie on the third side of the triangle and then proved this assumption wrong. To prove the theorem, we used one very common property of circles – angle in a semicircle is always a right angle.

Angle in a semicircle is a right angle

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.

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

Most of the questions in this exercise are based on the theorem – same segment angles are equal. We have also used the Isosceles triangle theorem in to solve this question along with the angle sum property of a triangle.

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

Theorem 7.2: Angles opposite to equal sides of an isosceles triangle are equal.

Theorem 6.7: The sum of the angles of a triangle is 180º.

Again, same concept – Angles in same segment are equal. Also used is the exterior angle theorem to solve this easy question. Below are the video tutorials that you may watch just in case you need to revise these concepts.

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

Exterior angle theorem: If a side of a triangle is produced, then the exterior angle so formed is equal to the sum of the two interior opposite angles.

Simple!

Concepts used:

• Angle sum property of triangle
• Angles in the same segment of a circle are equal

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

Theorem 6.7: The sum of the angles of a triangle is 180º.