 Hello all and welcome to the session. Today, the question is, A, B, C, and C, D are the three consecutive sites of a regular polygon. If angle BAC is equal to 15 degrees, find first part each interior angle of the polygon and second part number of sites of the polygon. Now, before starting the solution of the question, we should recall some of the results. The first result is, in a triangle, angles opposite to equal sites are equal to some property of a triangle which states that the sum of three angles of a triangle is 180 degrees. So, this angle sum property is a very important result which is going to be used in the given question. So, these results will be the key idea for the given question. And now, we will start with the solution. Let A, B, C, D and so on be a regular polygon. You can easily see in the diagram A, B, C, D and so on which we have taken as a regular polygon and let each interior angle of the polygon be x degrees. That is, each interior angle which means angle A, B, C is equal to x degrees which means angle A, B, C is equal to x degrees and given angle B, A, C is equal to 15 degrees. Now, we know that in a regular polygon, two sites therefore A, B is equal to B, C is equal to C, D is equal to so on. A, B is equal to B, C is equal to C, D and so on. Now, in triangle A, B, C, A, B is equal to B, C as these are a site of the regular polygon. So therefore, angle B, C is equal to angle B, C, A, angle B, C is equal to angle B, C, A because we know in a triangle two is opposite to equal size angle B, C, A is equal to 15 degrees because angle B, C is also 15 degrees. Therefore, B, C, A will also become 15 degrees. Now, in triangle A, B, C, angle A, B, C plus angle B, C plus angle B, C, A is equal to 180 degrees because of some angle property. This implies x degrees plus 15 degrees plus 15 degrees is equal to 180 degrees. This implies x degrees plus 30 degrees is equal to 180 degrees. This implies x degrees is equal to to 100 degrees minus 30 degrees. This implies x degrees is equal to 150 degrees. Now we know that a regular polygon with the angles equal therefore the interior angle of a polygon is equal to 150 degrees. So this is the solution of the first part and now we come to the solution of the second part of the question. Now for the second part we have to find out the number of sides of the polygon. Now we know that each interior angle of a regular polygon with n sides is equal to 2n-4 over n for right angles. Now from the first part each interior angle is equal to 150 degrees which will be equal to 2n-4 over n into 90 degrees. This implies 150 degrees over 90 degrees is equal to 2n-4 over n. Now here 0 will be cancelled with 0 and 3 into 3 is 9 and 3 into 5 is 15. So this implies 5 over 3 is equal to 2n-4 over n. Now cross multiplying we get 3 into 2n-4 is equal to 5 over n which implies 6n-12 is equal to 5n. This implies 6n-5n is equal to 12. This implies n is equal to 12. Therefore the number of sides of regular polygon is equal to 12. So this is the solution of the second part. That's all for this session. Hope you have enjoyed this session.